Explanation:
Okay, so I need to find the remainder when $129^{34} + 96^{38}$ is divided by 11. Hmm, let me think about how to approach this. I remember something about modular arithmetic and maybe Fermat's Little Theorem? Let me recall.

First, maybe I should simplify the bases 129 and 96 modulo 11. That way, instead of dealing with large numbers, I can work with smaller ones. Let me start with 129 divided by 11.

Let me calculate 11 times 11, which is 121. So 129 minus 121 is 8. Therefore, 129 is congruent to 8 modulo 11. So $129 \equiv 8 \mod 11$.

Now for 96. Let me divide 96 by 11. 11 times 8 is 88, so 96 minus 88 is 8. Wait, so 96 is also congruent to 8 modulo 11? Wait, that seems interesting. Let me check again. 11*8=88, 96-88=8. Yes, so 96 ≡ 8 mod 11. So both 129 and 96 are congruent to 8 modulo 11.

So the original expression $129^{34} + 96^{38}$ modulo 11 is equivalent to $8^{34} + 8^{38}$ modulo 11.

So now I need to compute $8^{34} + 8^{38} \mod 11$. Maybe I can factor out some common terms? Let me see.

First, note that $8^{34} + 8^{38} = 8^{34}(1 + 8^{4})$. Because $8^{38} = 8^{34} \times 8^4$. Let me check that exponent: 34 + 4 = 38, yes. So that's correct.

So now, I need to compute $8^{34} \times (1 + 8^4) \mod 11$.

But maybe there's a better way. Alternatively, since exponents can be simplified using Fermat's Little Theorem. Since 11 is prime, Fermat's Little Theorem says that for any integer a not divisible by 11, $a^{10} \equiv 1 \mod 11$.

So since 8 and 11 are coprime, we can use this theorem. Let me recall that.

So, $8^{10} \equiv 1 \mod 11$. Therefore, exponents can be reduced modulo 10.

So for $8^{34}$, let me divide 34 by 10. 10*3=30, so 34=10*3 + 4. Therefore, $8^{34} = (8^{10})^3 \times 8^4 \equiv 1^3 \times 8^4 = 8^4 \mod 11$.

Similarly, for $8^{38}$, divide 38 by 10. 10*3=30, 38-30=8. So 38=10*3 + 8. Therefore, $8^{38} = (8^{10})^3 \times 8^8 \equiv 1^3 \times 8^8 = 8^8 \mod 11$.

Therefore, the original expression becomes $8^4 + 8^8 \mod 11$.

So now I need to compute $8^4$ and $8^8$ modulo 11. Let me compute these step by step.

First, compute $8^1 \mod 11 = 8$.

$8^2 = 64$. Let me divide 64 by 11. 11*5=55, 64-55=9. So $8^2 \equiv 9 \mod 11$.

Then $8^3 = 8^2 * 8 = 9 * 8 = 72$. 72 divided by 11: 11*6=66, 72-66=6. So $8^3 \equiv 6 \mod 11$.

$8^4 = 8^3 * 8 = 6 * 8 = 48$. 48 divided by 11: 11*4=44, 48-44=4. So $8^4 \equiv 4 \mod 11$.

Okay, so $8^4 \equiv 4$. Now, $8^8$ is $(8^4)^2$. So since $8^4 \equiv 4$, then $8^8 \equiv 4^2 = 16 \mod 11$. 16 divided by 11 is 1 with remainder 5, so $8^8 \equiv 5 \mod 11$.

Therefore, $8^4 + 8^8 \equiv 4 + 5 = 9 \mod 11$.

Wait, so the total is 9? Let me check my steps again to make sure I didn't make a mistake.

First, simplifying 129 and 96 mod 11: 129 divided by 11 is 11*11=121, remainder 8. 96 divided by 11 is 8*11=88, remainder 8. So both are 8 mod 11. Correct.

Then, the expression becomes 8^34 + 8^38 mod 11. Then, using Fermat's theorem, since 8^10 ≡ 1 mod 11.

For 8^34: exponent 34 divided by 10 is 3 with remainder 4, so 8^34 ≡ 8^4 mod 11. Similarly, 8^38: exponent 38 divided by 10 is 3 with remainder 8, so 8^38 ≡ 8^8 mod 11.

Then, computing 8^4: 8^2 is 64 ≡ 9, 8^3 is 9*8=72≡6, 8^4 is 6*8=48≡4. Correct.

Then 8^8 is (8^4)^2 ≡ 4^2=16≡5 mod 11. Then 4 + 5 = 9. So the remainder is 9?

Wait, but let me check if there's another way to compute 8^8 mod 11. Maybe compute step by step?

Alternatively, since 8^4 ≡ 4, then 8^5 = 8^4 * 8 ≡ 4*8=32≡10 mod 11.

Then 8^6 = 8^5 * 8 ≡ 10*8=80≡80-77=3 mod 11.

8^7 = 3*8=24≡2 mod 11.

8^8 = 2*8=16≡5 mod 11.

Yes, same result. So 8^8 ≡5. So 4 + 5 = 9.

Therefore, the remainder is 9?

Wait, but let me check once more. Maybe I made a mistake in the exponents? Let me check 8^4 again.

8^1 = 8 mod 11.

8^2 = 64 mod 11. 11*5=55, 64-55=9. Correct.

8^3 = 8^2 * 8 = 9*8=72. 72 divided by 11: 11*6=66, 72-66=6. Correct.

8^4 = 6*8=48. 48-44=4. Correct.

So 8^4 is 4. Then 8^8 is 4^2=16≡5. Correct.

So 4 + 5 = 9. Therefore, the answer is 9?

But let me check with another approach. Maybe compute 8^34 + 8^38 mod 11.

Alternatively, since 8^10 ≡1 mod 11, then 8^34 = 8^(10*3 + 4) = (8^10)^3 * 8^4 ≡ 1^3 * 8^4 ≡ 8^4. Similarly, 8^38 = 8^(10*3 + 8) = (8^10)^3 * 8^8 ≡ 8^8. So same as before.

Alternatively, maybe compute exponents modulo 10? Wait, no, because Fermat's theorem says that exponents can be reduced modulo 10.

Alternatively, maybe compute 8^34 mod 11 and 8^38 mod 11 separately.

But I think my previous steps are correct.

Alternatively, maybe compute 8^4 + 8^8 mod 11.

But 8^4 is 4, 8^8 is 5, so 4 + 5 = 9.

Alternatively, maybe check with another exponent. Let me compute 8^10 mod 11.

Since 8^10 ≡1 mod 11. Let me check that.

We can compute 8^5 first. Earlier, we saw 8^5 ≡10 mod 11. Then 8^10 = (8^5)^2 ≡10^2=100≡1 mod 11. Because 100 divided by 11 is 9*11=99, remainder 1. Yes, so 8^10 ≡1 mod 11. Correct.

Therefore, the calculations are consistent.

Therefore, the answer is 9?

Wait, but let me check with another method. Maybe using negative numbers? Sometimes it's easier to work with negative residues.

Since 8 mod 11 is 8, but maybe 8 ≡ -3 mod 11? Because 11 - 3 = 8. So 8 ≡ -3 mod 11. Maybe that's easier for some exponents? Let me try.

So if I write 8 ≡ -3 mod 11. Then 8^4 ≡ (-3)^4 = 81 mod 11. 81 divided by 11: 11*7=77, 81-77=4. So same as before, 4.

Similarly, 8^8 ≡ (-3)^8 = 6561 mod 11. But maybe compute step by step.

(-3)^2 = 9 mod 11.

(-3)^4 = (9)^2 = 81 ≡4 mod 11.

(-3)^8 = (4)^2 = 16 ≡5 mod 11.

Same result. So 4 + 5 = 9.

So same answer.

Alternatively, maybe compute 8^34 + 8^38 mod 11. Let me think if there's another way.

Alternatively, factor 8^34(1 + 8^4). So 8^34*(1 + 8^4).

Earlier, we found that 8^4 ≡4, so 1 + 4 = 5. Therefore, the expression is 8^34 * 5 mod 11.

But 8^34 mod 11 is 8^4 mod 11, which is 4. Therefore, 4 * 5 = 20 mod 11. 20 - 11 = 9. So same result.

Therefore, regardless of the method, I get 9.

Wait, but let me check once more. Maybe I made a mistake in the initial step?

Original problem: 129^34 + 96^38 divided by 11.

129 mod 11: 129 / 11 = 11*11=121, remainder 8. Correct.

96 mod 11: 96 / 11 = 8*11=88, remainder 8. Correct.

So both bases are 8 mod 11. Therefore, expression is 8^34 + 8^38 mod 11.

Yes.

Alternatively, maybe compute 8^34 + 8^38 mod 11 as 8^34(1 + 8^4) mod 11.

Which is 8^34 * 5 mod 11.

But 8^34 mod 11: since 8^10 ≡1, so 8^34 = 8^(10*3 + 4) = (8^10)^3 * 8^4 ≡1^3 * 8^4 ≡8^4 ≡4.

So 4 * 5 = 20 ≡9 mod 11.

Same result.

Alternatively, maybe compute 8^34 + 8^38 mod 11 as follows:

Note that 8^34 + 8^38 = 8^34(1 + 8^4) = 8^34 * 5.

But 8^34 mod 11 is 8^(34 mod 10) = 8^4 mod 11? Wait, no. Wait, Fermat's theorem says that 8^10 ≡1, so exponents can be reduced modulo 10.

So 34 divided by 10 is 3 with remainder 4, so 8^34 ≡8^4. Similarly, 38 divided by 10 is 3 with remainder 8, so 8^38 ≡8^8.

But if I factor 8^34, then 8^34 * (1 + 8^4) is 8^34 * 5. But 8^34 is 8^4, so 8^4 * 5. Which is 4 * 5 = 20 ≡9.

Same result.

Alternatively, maybe compute 8^34 + 8^38 mod 11 as follows:

Let me compute 8^34 mod 11 and 8^38 mod 11 separately.

For 8^34:

As before, 34 divided by 10 is 3 with remainder 4, so 8^34 ≡8^4 ≡4.

For 8^38: 38 divided by 10 is 3 with remainder 8, so 8^38 ≡8^8 ≡5.

So 4 + 5 = 9.

Same answer.

Therefore, I think the answer is 9.

But just to be thorough, let me check with actual exponentiation.

Compute 8^4 mod 11: 8^4 = 4096. Let me divide 4096 by 11.

But that's tedious. Alternatively, since we already did step by step and got 4, and verified with negative numbers, maybe it's safe.

Alternatively, compute 8^4 mod 11:

8^1 = 8

8^2 = 64 mod 11: 64 - 5*11 = 64 - 55 = 9

8^3 = 8^2 * 8 = 9*8 = 72 mod 11: 72 - 6*11 = 72 - 66 = 6

8^4 = 6*8 = 48 mod 11: 48 - 4*11 = 48 - 44 = 4

Yes, so 4.

Similarly, 8^8:

8^4 = 4

8^5 = 4*8 = 32 mod 11: 32 - 2*11 = 10

8^6 = 10*8 = 80 mod 11: 80 - 7*11 = 80 - 77 = 3

8^7 = 3*8 = 24 mod 11: 24 - 2*11 = 2

8^8 = 2*8 = 16 mod 11: 16 - 11 = 5

Yes, so 5.

Therefore, 4 + 5 = 9.

Therefore, the remainder is 9.

But just to be 100% sure, let me check with another approach. Let me compute 8^10 ≡1 mod 11.

So 8^10 ≡1. Therefore, 8^10k ≡1^k=1.

So 8^34 = 8^(10*3 + 4) = (8^10)^3 * 8^4 ≡1^3 * 8^4 ≡8^4.

Similarly, 8^38 = 8^(10*3 + 8) = (8^10)^3 * 8^8 ≡1^3 * 8^8 ≡8^8.

So again, same result.

Alternatively, maybe compute exponents using exponents modulo 10?

But since 8^10 ≡1, then exponents can be reduced modulo 10.

So 34 mod 10 is 4, so 8^34 ≡8^4.

Similarly, 38 mod 10 is 8, so 8^38 ≡8^8.

Same as before.

Therefore, I think my answer is correct.

But just to make sure, let me check with smaller exponents.

For example, let me compute 8^1 + 8^1 mod 11. That would be 8 + 8 = 16 ≡5 mod 11.

Alternatively, if I use the same method: 8^1 + 8^1 = 8^1*(1 + 8^0) = 8*(1 + 1) = 8*2=16≡5. Correct.

Another test: 8^2 + 8^2 = 2*8^2 = 2*9=18≡7 mod 11.

Alternatively, compute directly: 64 + 64 = 128. 128 divided by 11: 11*11=121, 128-121=7. Correct.

So the method works.

Therefore, I think my answer is correct.

Therefore, the remainder when $129^{34} + 96^{38}$ is divided by 11 is 9.

**Final Answer**
The remainder is \boxed{9}.

To find the remainder when $129^{34} + 96^{38}$ is divided by 11, we can simplify the problem using **modular arithmetic** and **Fermat's Little Theorem**.

---

### Step 1: Reduce the Bases Modulo 11

We begin by reducing the bases $129$ and $96$ modulo 11:

- $129 \div 11 = 11 \times 11 + 8 \Rightarrow 129 \equiv 8 \mod 11$
- $96 \div 11 = 8 \times 11 + 8 \Rightarrow 96 \equiv 8 \mod 11$

So, both $129$ and $96$ are congruent to $8$ modulo 11. Therefore, the original expression becomes:

$$
129^{34} + 96^{38} \equiv 8^{34} + 8^{38} \mod 11
$$

---

### Step 2: Apply Fermat's Little Theorem

Since 11 is a prime number, **Fermat's Little Theorem** tells us that for any integer $a$ not divisible by 11:

$$
a^{10} \equiv 1 \mod 11
$$

This allows us to reduce the exponents modulo 10:

- $34 \mod 10 = 4 \Rightarrow 8^{34} \equiv 8^4 \mod 11$
- $38 \mod 10 = 8 \Rightarrow 8^{38} \equiv 8^8 \mod 11$

So the expression becomes:

$$
8^{34} + 8^{38} \equiv 8^4 + 8^8 \mod 11
$$

---

### Step 3: Compute Powers of 8 Modulo 11

We compute the powers of 8 modulo 11 step by step:

- $8^1 \equiv 8 \mod 11$
- $8^2 = 64 \equiv 9 \mod 11$
- $8^3 = 8^2 \cdot 8 = 9 \cdot 8 = 72 \equiv 6 \mod 11$
- $8^4 = 8^3 \cdot 8 = 6 \cdot 8 = 48 \equiv 4 \mod 11$
- $8^5 = 8^4 \cdot 8 = 4 \cdot 8 = 32 \equiv 10 \mod 11$
- $8^6 = 8^5 \cdot 8 = 10 \cdot 8 = 80 \equiv 3 \mod 11$
- $8^7 = 8^6 \cdot 8 = 3 \cdot 8 = 24 \equiv 2 \mod 11$
- $8^8 = 8^7 \cdot 8 = 2 \cdot 8 = 16 \equiv 5 \mod 11$

So:

- $8^4 \equiv 4 \mod 11$
- $8^8 \equiv 5 \mod 11$

---

### Step 4: Add the Results

Now we compute:

$$
8^4 + 8^8 \equiv 4 + 5 = 9 \mod 11
$$

---

### Final Answer

$$
\boxed{9}
$$

[Assistant]
Explanation:
Okay, so I need to find the remainder when $129^{34} + 96^{38}$ is divided by 11. Hmm, let me think about how to approach this. I remember something about modular arithmetic and maybe Fermat's Little Theorem? Let me recall.

First, maybe I should simplify the bases 129 and 96 modulo 11. That way, instead of dealing with large numbers, I can work with smaller ones. Let me start with 129 divided by 11.

Let me calculate 11 times 11, which is 121. So 129 minus 121 is 8. Therefore, 129 is congruent to 8 modulo 11. So $129 \equiv 8 \mod 11$.

Now for 96. Let me divide 96 by 11. 11 times 8 is 88, so 96 minus 88 is 8. Wait, so 96 is also congruent to 8 modulo 11? Wait, that seems interesting. Let me check again. 11*8=88, 96-88=8. Yes, so 96 ≡ 8 mod 11. So both 129 and 96 are congruent to 8 modulo 11.

So the original expression $129^{34} + 96^{38}$ modulo 11 is equivalent to $8^{34} + 8^{38}$ modulo 11.

So now I need to compute $8^{34} + 8^{38} \mod 11$. Maybe I can factor out some common terms? Let me see.

First, note that $8^{34} + 8^{38} = 8^{34}(1 + 8^{4})$. Because $8^{38} = 8^{34} \times 8^4$. Let me check that exponent: 34 + 4 = 38, yes. So that's correct.

So now, I need to compute $8^{34} \times (1 + 8^4) \mod 11$.

But maybe there's a better way. Alternatively, since exponents can be simplified using Fermat's Little Theorem. Since 11 is prime, Fermat's Little Theorem says that for any integer a not divisible by 11, $a^{10} \equiv 1 \mod 11$.

So since 8 and 11 are coprime, we can use this theorem. Let me recall that.

So, $8^{10} \equiv 1 \mod 11$. Therefore, exponents can be reduced modulo 10.

So for $8^{34}$, let me divide 34 by 10. 10*3=30, so 34=10*3 + 4. Therefore, $8^{34} = (8^{10})^3 \times 8^4 \equiv 1^3 \times 8^4 = 8^4 \mod 11$.

Similarly, for $8^{38}$, divide 38 by 10. 10*3=30, 38-30=8. So 38=10*3 + 8. Therefore, $8^{38} = (8^{10})^3 \times 8^8 \equiv 1^3 \times 8^8 = 8^8 \mod 11$.

Therefore, the original expression becomes $8^4 + 8^8 \mod 11$.

So now I need to compute $8^4$ and $8^8$ modulo 11. Let me compute these step by step.

First, compute $8^1 \mod 11 = 8$.

$8^2 = 64$. Let me divide 64 by 11. 11*5=55, 64-55=9. So $8^2 \equiv 9 \mod 11$.

Then $8^3 = 8^2 * 8 = 9 * 8 = 72$. 72 divided by 11: 11*6=66, 72-66=6. So $8^3 \equiv 6 \mod 11$.

$8^4 = 8^3 * 8 = 6 * 8 = 48$. 48 divided by 11: 11*4=44, 48-44=4. So $8^4 \equiv 4 \mod 11$.

Okay, so $8^4 \equiv 4$. Now, $8^8$ is $(8^4)^2$. So since $8^4 \equiv 4$, then $8^8 \equiv 4^2 = 16 \mod 11$. 16 divided by 11 is 1 with remainder 5, so $8^8 \equiv 5 \mod 11$.

Therefore, $8^4 + 8^8 \equiv 4 + 5 = 9 \mod 11$.

Wait, so the total is 9? Let me check my steps again to make sure I didn't make a mistake.

First, simplifying 129 and 96 mod 11: 129 divided by 11 is 11*11=121, remainder 8. 96 divided by 11 is 8*11=88, remainder 8. So both are 8 mod 11. Correct.

Then, the expression becomes 8^34 + 8^38 mod 11. Then, using Fermat's theorem, since 8^10 ≡ 1 mod 11.

For 8^34: exponent 34 divided by 10 is 3 with remainder 4, so 8^34 ≡ 8^4 mod 11. Similarly, 8^38: exponent 38 divided by 10 is 3 with remainder 8, so 8^38 ≡ 8^8 mod 11.

Then, computing 8^4: 8^2 is 64 ≡ 9, 8^3 is 9*8=72≡6, 8^4 is 6*8=48≡4. Correct.

Then 8^8 is (8^4)^2 ≡ 4^2=16≡5 mod 11. Then 4 + 5 = 9. So the remainder is 9?

Wait, but let me check if there's another way to compute 8^8 mod 11. Maybe compute step by step?

Alternatively, since 8^4 ≡ 4, then 8^5 = 8^4 * 8 ≡ 4*8=32≡10 mod 11.

Then 8^6 = 8^5 * 8 ≡ 10*8=80≡80-77=3 mod 11.

8^7 = 3*8=24≡2 mod 11.

8^8 = 2*8=16≡5 mod 11.

Yes, same result. So 8^8 ≡5. So 4 + 5 = 9.

Therefore, the remainder is 9?

Wait, but let me check once more. Maybe I made a mistake in the exponents? Let me check 8^4 again.

8^1 = 8 mod 11.

8^2 = 64 mod 11. 11*5=55, 64-55=9. Correct.

8^3 = 8^2 * 8 = 9*8=72. 72 divided by 11: 11*6=66, 72-66=6. Correct.

8^4 = 6*8=48. 48-44=4. Correct.

So 8^4 is 4. Then 8^8 is 4^2=16≡5. Correct.

So 4 + 5 = 9. Therefore, the answer is 9?

But let me check with another approach. Maybe compute 8^34 + 8^38 mod 11.

Alternatively, since 8^10 ≡1 mod 11, then 8^34 = 8^(10*3 + 4) = (8^10)^3 * 8^4 ≡ 1^3 * 8^4 ≡ 8^4. Similarly, 8^38 = 8^(10*3 + 8) = (8^10)^3 * 8^8 ≡ 8^8. So same as before.

Alternatively, maybe compute exponents modulo 10? Wait, no, because Fermat's theorem says that exponents can be reduced modulo 10.

Alternatively, maybe compute 8^34 mod 11 and 8^38 mod 11 separately.

But I think my previous steps are correct.

Alternatively, maybe compute 8^4 + 8^8 mod 11.

But 8^4 is 4, 8^8 is 5, so 4 + 5 = 9.

Alternatively, maybe check with another exponent. Let me compute 8^10 mod 11.

Since 8^10 ≡1 mod 11. Let me check that.

We can compute 8^5 first. Earlier, we saw 8^5 ≡10 mod 11. Then 8^10 = (8^5)^2 ≡10^2=100≡1 mod 11. Because 100 divided by 11 is 9*11=99, remainder 1. Yes, so 8^10 ≡1 mod 11. Correct.

Therefore, the calculations are consistent.

Therefore, the answer is 9?

Wait, but let me check with another method. Maybe using negative numbers? Sometimes it's easier to work with negative residues.

Since 8 mod 11 is 8, but maybe 8 ≡ -3 mod 11? Because 11 - 3 = 8. So 8 ≡ -3 mod 11. Maybe that's easier for some exponents? Let me try.

So if I write 8 ≡ -3 mod 11. Then 8^4 ≡ (-3)^4 = 81 mod 11. 81 divided by 11: 11*7=77, 81-77=4. So same as before, 4.

Similarly, 8^8 ≡ (-3)^8 = 6561 mod 11. But maybe compute step by step.

(-3)^2 = 9 mod 11.

(-3)^4 = (9)^2 = 81 ≡4 mod 11.

(-3)^8 = (4)^2 = 16 ≡5 mod 11.

Same result. So 4 + 5 = 9.

So same answer.

Alternatively, maybe compute 8^34 + 8^38 mod 11. Let me think if there's another way.

Alternatively, factor 8^34(1 + 8^4). So 8^34*(1 + 8^4).

Earlier, we found that 8^4 ≡4, so 1 + 4 = 5. Therefore, the expression is 8^34 * 5 mod 11.

But 8^34 mod 11 is 8^4 mod 11, which is 4. Therefore, 4 * 5 = 20 mod 11. 20 - 11 = 9. So same result.

Therefore, regardless of the method, I get 9.

Wait, but let me check once more. Maybe I made a mistake in the initial step?

Original problem: 129^34 + 96^38 divided by 11.

129 mod 11: 129 / 11 = 11*11=121, remainder 8. Correct.

96 mod 11: 96 / 11 = 8*11=88, remainder 8. Correct.

So both bases are 8 mod 11. Therefore, expression is 8^34 + 8^38 mod 11.

Yes.

Alternatively, maybe compute 8^34 + 8^38 mod 11 as 8^34(1 + 8^4) mod 11.

Which is 8^34 * 5 mod 11.

But 8^34 mod 11: since 8^10 ≡1, so 8^34 = 8^(10*3 + 4) = (8^10)^3 * 8^4 ≡1^3 * 8^4 ≡8^4 ≡4.

So 4 * 5 = 20 ≡9 mod 11.

Same result.

Alternatively, maybe compute 8^34 + 8^38 mod 11 as follows:

Note that 8^34 + 8^38 = 8^34(1 + 8^4) = 8^34 * 5.

But 8^34 mod 11 is 8^(34 mod 10) = 8^4 mod 11? Wait, no. Wait, Fermat's theorem says that 8^10 ≡1, so exponents can be reduced modulo 10.

So 34 divided by 10 is 3 with remainder 4, so 8^34 ≡8^4. Similarly, 38 divided by 10 is 3 with remainder 8, so 8^38 ≡8^8.

But if I factor 8^34, then 8^34 * (1 + 8^4) is 8^34 * 5. But 8^34 is 8^4, so 8^4 * 5. Which is 4 * 5 = 20 ≡9.

Same result.

Alternatively, maybe compute 8^34 + 8^38 mod 11 as follows:

Let me compute 8^34 mod 11 and 8^38 mod 11 separately.

For 8^34:

As before, 34 divided by 10 is 3 with remainder 4, so 8^34 ≡8^4 ≡4.

For 8^38: 38 divided by 10 is 3 with remainder 8, so 8^38 ≡8^8 ≡5.

So 4 + 5 = 9.

Same answer.

Therefore, I think the answer is 9.

But just to be thorough, let me check with actual exponentiation.

Compute 8^4 mod 11: 8^4 = 4096. Let me divide 4096 by 11.

But that's tedious. Alternatively, since we already did step by step and got 4, and verified with negative numbers, maybe it's safe.

Alternatively, compute 8^4 mod 11:

8^1 = 8

8^2 = 64 mod 11: 64 - 5*11 = 64 - 55 = 9

8^3 = 8^2 * 8 = 9*8 = 72 mod 11: 72 - 6*11 = 72 - 66 = 6

8^4 = 6*8 = 48 mod 11: 48 - 4*11 = 48 - 44 = 4

Yes, so 4.

Similarly, 8^8:

8^4 = 4

8^5 = 4*8 = 32 mod 11: 32 - 2*11 = 10

8^6 = 10*8 = 80 mod 11: 80 - 7*11 = 80 - 77 = 3

8^7 = 3*8 = 24 mod 11: 24 - 2*11 = 2

8^8 = 2*8 = 16 mod 11: 16 - 11 = 5

Yes, so 5.

Therefore, 4 + 5 = 9.

Therefore, the remainder is 9.

But just to be 100% sure, let me check with another approach. Let me compute 8^10 ≡1 mod 11.

So 8^10 ≡1. Therefore, 8^10k ≡1^k=1.

So 8^34 = 8^(10*3 + 4) = (8^10)^3 * 8^4 ≡1^3 * 8^4 ≡8^4.

Similarly, 8^38 = 8^(10*3 + 8) = (8^10)^3 * 8^8 ≡1^3 * 8^8 ≡8^8.

So again, same result.

Alternatively, maybe compute exponents using exponents modulo 10?

But since 8^10 ≡1, then exponents can be reduced modulo 10.

So 34 mod 10 is 4, so 8^34 ≡8^4.

Similarly, 38 mod 10 is 8, so 8^38 ≡8^8.

Same as before.

Therefore, I think my answer is correct.

But just to make sure, let me check with smaller exponents.

For example, let me compute 8^1 + 8^1 mod 11. That would be 8 + 8 = 16 ≡5 mod 11.

Alternatively, if I use the same method: 8^1 + 8^1 = 8^1*(1 + 8^0) = 8*(1 + 1) = 8*2=16≡5. Correct.

Another test: 8^2 + 8^2 = 2*8^2 = 2*9=18≡7 mod 11.

Alternatively, compute directly: 64 + 64 = 128. 128 divided by 11: 11*11=121, 128-121=7. Correct.

So the method works.

Therefore, I think my answer is correct.

Therefore, the remainder when $129^{34} + 96^{38}$ is divided by 11 is 9.

**Final Answer**
The remainder is \boxed{9}.

To find the remainder when $129^{34} + 96^{38}$ is divided by 11, we can simplify the problem using **modular arithmetic** and **Fermat's Little Theorem**.

---

### Step 1: Reduce the Bases Modulo 11

We begin by reducing the bases $129$ and $96$ modulo 11:

- $129 \div 11 = 11 \times 11 + 8 \Rightarrow 129 \equiv 8 \mod 11$
- $96 \div 11 = 8 \times 11 + 8 \Rightarrow 96 \equiv 8 \mod 11$

So, both $129$ and $96$ are congruent to $8$ modulo 11. Therefore, the original expression becomes:

$$
129^{34} + 96^{38} \equiv 8^{34} + 8^{38} \mod 11
$$

---

### Step 2: Apply Fermat's Little Theorem

Since 11 is a prime number, **Fermat's Little Theorem** tells us that for any integer $a$ not divisible by 11:

$$
a^{10} \equiv 1 \mod 11
$$

This allows us to reduce the exponents modulo 10:

- $34 \mod 10 = 4 \Rightarrow 8^{34} \equiv 8^4 \mod 11$
- $38 \mod 10 = 8 \Rightarrow 8^{38} \equiv 8^8 \mod 11$

So the expression becomes:

$$
8^{34} + 8^{38} \equiv 8^4 + 8^8 \mod 11
$$

---

### Step 3: Compute Powers of 8 Modulo 11

We compute the powers of 8 modulo 11 step by step:

- $8^1 \equiv 8 \mod 11$
- $8^2 = 64 \equiv 9 \mod 11$
- $8^3 = 8^2 \cdot 8 = 9 \cdot 8 = 72 \equiv 6 \mod 11$
- $8^4 = 8^3 \cdot 8 = 6 \cdot 8 = 48 \equiv 4 \mod 11$
- $8^5 = 8^4 \cdot 8 = 4 \cdot 8 = 32 \equiv 10 \mod 11$
- $8^6 = 8^5 \cdot 8 = 10 \cdot 8 = 80 \equiv 3 \mod 11$
- $8^7 = 8^6 \cdot 8 = 3 \cdot 8 = 24 \equiv 2 \mod 11$
- $8^8 = 8^7 \cdot 8 = 2 \cdot 8 = 16 \equiv 5 \mod 11$

So:

- $8^4 \equiv 4 \mod 11$
- $8^8 \equiv 5 \mod 11$

---

### Step 4: Add the Results

Now we compute:

$$
8^4 + 8^8 \equiv 4 + 5 = 9 \mod 11
$$

---

### Final Answer

$$
\boxed{9}
$$

Solution:
We use the property that $a \equiv b \pmod{m}$ implies $a^c \equiv b^c \pmod{m}$.

Since $129 \equiv -3 \pmod{11}$ and $96 \equiv -3 \pmod{11}$, we have  $$129^{34}+96^{38} \equiv (-3)^{34}+(-3)^{38} \equiv 3^{34}+3^{38} \pmod{11}.$$Since $3^5 \equiv 1 \pmod{11},$ we can see that $3^{34} = (3^5)^{6} \cdot 3^4$ and $3^{38} = (3^5)^{7} \cdot 3^3.$

Then,  \begin{align*}
129^{34}+96^{38}&\equiv (3^5)^{6} \cdot 3^4 + (3^5)^{7} \cdot 3^3\\
& \equiv 3^4 + 3^3\\
& \equiv 81 + 27\\
& \equiv 108 \\
&\equiv \boxed{9} \pmod{11}.
\end{align*}


Explanation:
Okay, so I need to find the sum of the coefficients of the polynomial g(x), which is f(x - 1). The original polynomial f(x) is given as 3x⁴ + 5x² - 9x - 2. Hmm, let me recall how to approach this.

First, I remember that the sum of the coefficients of a polynomial can be found by evaluating the polynomial at x = 1. For example, if you have a polynomial like ax² + bx + c, the sum of the coefficients is a + b + c, which is the same as plugging in x = 1: a(1)² + b(1) + c = a + b + c. So, applying that here, the sum of the coefficients of g(x) would be g(1).

But wait, g(x) is f(x - 1). So, g(1) would be f(1 - 1) = f(0). Therefore, the sum of the coefficients of g(x) is f(0). Is that right? Let me check.

Let me think again. If g(x) = f(x - 1), then substituting x = 1 into g(x) gives g(1) = f(0). But the sum of the coefficients of g(x) is indeed g(1). Therefore, the answer is f(0). Let me compute f(0).

Looking at f(x) = 3x⁴ + 5x² - 9x - 2. Plugging in x = 0: 3*(0)^4 + 5*(0)^2 - 9*(0) - 2 = 0 + 0 - 0 - 2 = -2. So, is the answer -2? Wait, that seems straightforward, but maybe I need to verify this another way to make sure I didn't make a mistake.

Alternatively, maybe I should actually compute g(x) by expanding f(x - 1) and then find the sum of its coefficients. Let me try that method to confirm.

First, let's write f(x - 1). The original polynomial is 3x⁴ + 5x² - 9x - 2. So replacing every x with (x - 1):

f(x - 1) = 3(x - 1)^4 + 5(x - 1)^2 - 9(x - 1) - 2.

Now, I need to expand each term step by step.

Starting with (x - 1)^4. Let me recall the binomial expansion. (a - b)^4 = a^4 - 4a³b + 6a²b² - 4ab³ + b^4. So, (x - 1)^4 = x^4 - 4x³ + 6x² - 4x + 1.

Then, multiplying by 3: 3x^4 - 12x³ + 18x² - 12x + 3.

Next term is 5(x - 1)^2. Let's compute (x - 1)^2 first: x² - 2x + 1. Multiply by 5: 5x² - 10x + 5.

Third term is -9(x - 1): -9x + 9.

Fourth term is -2.

Now, let's combine all these terms:

First term: 3x^4 - 12x³ + 18x² - 12x + 3

Second term: + 5x² - 10x + 5

Third term: -9x + 9

Fourth term: -2

Now, let's add them all together term by term.

Starting with the x^4 term: 3x^4.

Next, x³ term: -12x³.

x² terms: 18x² + 5x² = 23x².

x terms: -12x -10x -9x = (-12 -10 -9)x = -31x.

Constant terms: 3 + 5 + 9 - 2 = (3 + 5) + (9 - 2) = 8 + 7 = 15.

So putting it all together, g(x) = 3x^4 - 12x³ + 23x² - 31x + 15.

Now, the sum of the coefficients is obtained by plugging in x = 1:

g(1) = 3(1)^4 - 12(1)^3 + 23(1)^2 - 31(1) + 15 = 3 - 12 + 23 - 31 + 15.

Let me compute that step by step:

Start with 3 - 12 = -9.

-9 + 23 = 14.

14 - 31 = -17.

-17 + 15 = -2.

So, the sum of the coefficients is indeed -2. Which matches the previous result where f(0) = -2. Therefore, both methods give the same answer, so I can be confident that the answer is -2.

But just to make sure I didn't make any mistakes in expanding, let me check the expansion again.

Original f(x - 1):

First term: 3(x - 1)^4. As above, that's 3x^4 - 12x³ + 18x² - 12x + 3. Correct.

Second term: 5(x - 1)^2 = 5(x² - 2x + 1) = 5x² - 10x + 5. Correct.

Third term: -9(x - 1) = -9x + 9. Correct.

Fourth term: -2. Correct.

Adding all together:

x^4: 3x^4.

x³: -12x³.

x²: 18x² + 5x² = 23x². Correct.

x terms: -12x -10x -9x. Let me check again: from first term: -12x, second term: -10x, third term: -9x. So total is -12 -10 -9 = -31x. Correct.

Constants: 3 (from first term) + 5 (from second term) + 9 (from third term) - 2 (from fourth term). 3 + 5 is 8, 8 + 9 is 17, 17 - 2 is 15. Correct. So the polynomial is correct.

Then evaluating at x = 1: 3 -12 +23 -31 +15. Let me compute again:

3 -12 is -9. -9 +23 is 14. 14 -31 is -17. -17 +15 is -2. Correct.

Alternatively, if I think about the original f(0) = -2. Since g(1) = f(0), which is -2. So both methods confirm that.

Therefore, the answer is -2. But let me think again if there's another way someone might approach this problem and maybe get confused.

Another way: sometimes people think that shifting the polynomial affects the coefficients in some way, but in reality, the sum of coefficients is just the value at x=1. So regardless of how the polynomial is transformed, if you want the sum of coefficients of g(x), it's g(1). Since g(x) = f(x - 1), then g(1) = f(0). Therefore, the answer is f(0). Which is straightforward. So maybe that's the quickest way. But since I was unsure, I expanded it out to check.

Alternatively, if someone didn't remember that trick, they might have to expand. But since both methods give the same answer, I think it's safe.

Therefore, the sum of the coefficients of g(x) is -2.

**Final Answer**
The sum of the coefficients of $ g $ is \boxed{-2}.

To find the sum of the coefficients of the polynomial $ g(x) = f(x - 1) $, we begin by recalling a key property of polynomials:

> The **sum of the coefficients** of a polynomial $ p(x) $ is equal to $ p(1) $.

---

### Step 1: Understand the Relationship Between $ f(x) $ and $ g(x) $

We are given:
$$
f(x) = 3x^4 + 5x^2 - 9x - 2
$$
and
$$
g(x) = f(x - 1)
$$

So, $ g(x) $ is a horizontal shift of $ f(x) $ to the right by 1 unit. However, for the purpose of finding the **sum of the coefficients**, we don't need to expand $ g(x) $ fully. Instead, we can use the fact that:

$$
\text{Sum of coefficients of } g(x) = g(1) = f(1 - 1) = f(0)
$$

---

### Step 2: Evaluate $ f(0) $

$$
f(0) = 3(0)^4 + 5(0)^2 - 9(0) - 2 = 0 + 0 - 0 - 2 = -2
$$

---

### Step 3: Confirm by Expanding $ g(x) $

To double-check, we can expand $ f(x - 1) $:

$$
f(x - 1) = 3(x - 1)^4 + 5(x - 1)^2 - 9(x - 1) - 2
$$

Compute each term:

- $ (x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1 $
- $ 3(x - 1)^4 = 3x^4 - 12x^3 + 18x^2 - 12x + 3 $
- $ (x - 1)^2 = x^2 - 2x + 1 $
- $ 5(x - 1)^2 = 5x^2 - 10x + 5 $
- $ -9(x - 1) = -9x + 9 $
- Constant term: $ -2 $

Now combine all terms:

$$
g(x) = 3x^4 - 12x^3 + 18x^2 - 12x + 3 + 5x^2 - 10x + 5 - 9x + 9 - 2
$$

Group like terms:

- $ x^4 $: $ 3x^4 $
- $ x^3 $: $ -12x^3 $
- $ x^2 $: $ 18x^2 + 5x^2 = 23x^2 $
- $ x $: $ -12x - 10x - 9x = -31x $
- Constants: $ 3 + 5 + 9 - 2 = 15 $

So,
$$
g(x) = 3x^4 - 12x^3 + 23x^2 - 31x + 15
$$

Now compute the sum of the coefficients by evaluating $ g(1) $:

$$
g(1) = 3(1)^4 - 12(1)^3 + 23(1)^2 - 31(1) + 15 = 3 - 12 + 23 - 31 + 15 = -2
$$

---

### Final Answer

$$
\boxed{-2}
$$

[Assistant]
Explanation:
Okay, so I need to find the sum of the coefficients of the polynomial g(x), which is f(x - 1). The original polynomial f(x) is given as 3x⁴ + 5x² - 9x - 2. Hmm, let me recall how to approach this.

First, I remember that the sum of the coefficients of a polynomial can be found by evaluating the polynomial at x = 1. For example, if you have a polynomial like ax² + bx + c, the sum of the coefficients is a + b + c, which is the same as plugging in x = 1: a(1)² + b(1) + c = a + b + c. So, applying that here, the sum of the coefficients of g(x) would be g(1).

But wait, g(x) is f(x - 1). So, g(1) would be f(1 - 1) = f(0). Therefore, the sum of the coefficients of g(x) is f(0). Is that right? Let me check.

Let me think again. If g(x) = f(x - 1), then substituting x = 1 into g(x) gives g(1) = f(0). But the sum of the coefficients of g(x) is indeed g(1). Therefore, the answer is f(0). Let me compute f(0).

Looking at f(x) = 3x⁴ + 5x² - 9x - 2. Plugging in x = 0: 3*(0)^4 + 5*(0)^2 - 9*(0) - 2 = 0 + 0 - 0 - 2 = -2. So, is the answer -2? Wait, that seems straightforward, but maybe I need to verify this another way to make sure I didn't make a mistake.

Alternatively, maybe I should actually compute g(x) by expanding f(x - 1) and then find the sum of its coefficients. Let me try that method to confirm.

First, let's write f(x - 1). The original polynomial is 3x⁴ + 5x² - 9x - 2. So replacing every x with (x - 1):

f(x - 1) = 3(x - 1)^4 + 5(x - 1)^2 - 9(x - 1) - 2.

Now, I need to expand each term step by step.

Starting with (x - 1)^4. Let me recall the binomial expansion. (a - b)^4 = a^4 - 4a³b + 6a²b² - 4ab³ + b^4. So, (x - 1)^4 = x^4 - 4x³ + 6x² - 4x + 1.

Then, multiplying by 3: 3x^4 - 12x³ + 18x² - 12x + 3.

Next term is 5(x - 1)^2. Let's compute (x - 1)^2 first: x² - 2x + 1. Multiply by 5: 5x² - 10x + 5.

Third term is -9(x - 1): -9x + 9.

Fourth term is -2.

Now, let's combine all these terms:

First term: 3x^4 - 12x³ + 18x² - 12x + 3

Second term: + 5x² - 10x + 5

Third term: -9x + 9

Fourth term: -2

Now, let's add them all together term by term.

Starting with the x^4 term: 3x^4.

Next, x³ term: -12x³.

x² terms: 18x² + 5x² = 23x².

x terms: -12x -10x -9x = (-12 -10 -9)x = -31x.

Constant terms: 3 + 5 + 9 - 2 = (3 + 5) + (9 - 2) = 8 + 7 = 15.

So putting it all together, g(x) = 3x^4 - 12x³ + 23x² - 31x + 15.

Now, the sum of the coefficients is obtained by plugging in x = 1:

g(1) = 3(1)^4 - 12(1)^3 + 23(1)^2 - 31(1) + 15 = 3 - 12 + 23 - 31 + 15.

Let me compute that step by step:

Start with 3 - 12 = -9.

-9 + 23 = 14.

14 - 31 = -17.

-17 + 15 = -2.

So, the sum of the coefficients is indeed -2. Which matches the previous result where f(0) = -2. Therefore, both methods give the same answer, so I can be confident that the answer is -2.

But just to make sure I didn't make any mistakes in expanding, let me check the expansion again.

Original f(x - 1):

First term: 3(x - 1)^4. As above, that's 3x^4 - 12x³ + 18x² - 12x + 3. Correct.

Second term: 5(x - 1)^2 = 5(x² - 2x + 1) = 5x² - 10x + 5. Correct.

Third term: -9(x - 1) = -9x + 9. Correct.

Fourth term: -2. Correct.

Adding all together:

x^4: 3x^4.

x³: -12x³.

x²: 18x² + 5x² = 23x². Correct.

x terms: -12x -10x -9x. Let me check again: from first term: -12x, second term: -10x, third term: -9x. So total is -12 -10 -9 = -31x. Correct.

Constants: 3 (from first term) + 5 (from second term) + 9 (from third term) - 2 (from fourth term). 3 + 5 is 8, 8 + 9 is 17, 17 - 2 is 15. Correct. So the polynomial is correct.

Then evaluating at x = 1: 3 -12 +23 -31 +15. Let me compute again:

3 -12 is -9. -9 +23 is 14. 14 -31 is -17. -17 +15 is -2. Correct.

Alternatively, if I think about the original f(0) = -2. Since g(1) = f(0), which is -2. So both methods confirm that.

Therefore, the answer is -2. But let me think again if there's another way someone might approach this problem and maybe get confused.

Another way: sometimes people think that shifting the polynomial affects the coefficients in some way, but in reality, the sum of coefficients is just the value at x=1. So regardless of how the polynomial is transformed, if you want the sum of coefficients of g(x), it's g(1). Since g(x) = f(x - 1), then g(1) = f(0). Therefore, the answer is f(0). Which is straightforward. So maybe that's the quickest way. But since I was unsure, I expanded it out to check.

Alternatively, if someone didn't remember that trick, they might have to expand. But since both methods give the same answer, I think it's safe.

Therefore, the sum of the coefficients of g(x) is -2.

**Final Answer**
The sum of the coefficients of $ g $ is \boxed{-2}.

To find the sum of the coefficients of the polynomial $ g(x) = f(x - 1) $, we begin by recalling a key property of polynomials:

> The **sum of the coefficients** of a polynomial $ p(x) $ is equal to $ p(1) $.

---

### Step 1: Understand the Relationship Between $ f(x) $ and $ g(x) $

We are given:
$$
f(x) = 3x^4 + 5x^2 - 9x - 2
$$
and
$$
g(x) = f(x - 1)
$$

So, $ g(x) $ is a horizontal shift of $ f(x) $ to the right by 1 unit. However, for the purpose of finding the **sum of the coefficients**, we don't need to expand $ g(x) $ fully. Instead, we can use the fact that:

$$
\text{Sum of coefficients of } g(x) = g(1) = f(1 - 1) = f(0)
$$

---

### Step 2: Evaluate $ f(0) $

$$
f(0) = 3(0)^4 + 5(0)^2 - 9(0) - 2 = 0 + 0 - 0 - 2 = -2
$$

---

### Step 3: Confirm by Expanding $ g(x) $

To double-check, we can expand $ f(x - 1) $:

$$
f(x - 1) = 3(x - 1)^4 + 5(x - 1)^2 - 9(x - 1) - 2
$$

Compute each term:

- $ (x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1 $
- $ 3(x - 1)^4 = 3x^4 - 12x^3 + 18x^2 - 12x + 3 $
- $ (x - 1)^2 = x^2 - 2x + 1 $
- $ 5(x - 1)^2 = 5x^2 - 10x + 5 $
- $ -9(x - 1) = -9x + 9 $
- Constant term: $ -2 $

Now combine all terms:

$$
g(x) = 3x^4 - 12x^3 + 18x^2 - 12x + 3 + 5x^2 - 10x + 5 - 9x + 9 - 2
$$

Group like terms:

- $ x^4 $: $ 3x^4 $
- $ x^3 $: $ -12x^3 $
- $ x^2 $: $ 18x^2 + 5x^2 = 23x^2 $
- $ x $: $ -12x - 10x - 9x = -31x $
- Constants: $ 3 + 5 + 9 - 2 = 15 $

So,
$$
g(x) = 3x^4 - 12x^3 + 23x^2 - 31x + 15
$$

Now compute the sum of the coefficients by evaluating $ g(1) $:

$$
g(1) = 3(1)^4 - 12(1)^3 + 23(1)^2 - 31(1) + 15 = 3 - 12 + 23 - 31 + 15 = -2
$$

---

### Final Answer

$$
\boxed{-2}
$$

Solution: The sum of the coefficients of $g(x)$ can be found by evaluating $g(1)$.  Since $g(x)=f(x-1)$, we know that $g(1)=f(1-1)=f(0)$.  Therefore the sum of the coefficients is equal to $f(0)=\boxed{-2}$.


Explanation:
Okay, so I need to find the shortest distance from the point (6, 0) to the line y = 2x - 2. Hmm, let me think. I remember that the shortest distance from a point to a line is the perpendicular distance. So maybe I can use the formula for the distance from a point to a line? Wait, what's that formula again?

I think it's something like |Ax + By + C| divided by the square root of A squared plus B squared. But let me make sure. The general formula for the distance from a point (x₀, y₀) to the line Ax + By + C = 0 is |Ax₀ + By₀ + C| / sqrt(A² + B²). Right, that sounds familiar. So first, I need to write the given line equation in the standard form Ax + By + C = 0.

The given line is y = 2x - 2. Let me rearrange that. Subtract 2x from both sides: -2x + y + 2 = 0. Or maybe I can write it as 2x - y - 2 = 0. Wait, let me check. If I move all terms to the left side: y - 2x + 2 = 0? Wait, no. Let me do it step by step.

Original equation: y = 2x - 2. Subtract 2x from both sides: -2x + y = -2. Then add 2 to both sides: -2x + y + 2 = 0. So that's the standard form. Alternatively, multiplying both sides by -1 to make the coefficient of x positive: 2x - y - 2 = 0. Either way, both forms are acceptable. Let me pick one. Let me go with 2x - y - 2 = 0. So A is 2, B is -1, and C is -2.

Wait, but in the formula, it's Ax + By + C = 0. So if I have 2x - y - 2 = 0, then A = 2, B = -1, C = -2. So then plugging into the distance formula: |A*x₀ + B*y₀ + C| / sqrt(A² + B²). Let me check that again. So the point is (6, 0). So x₀ is 6, y₀ is 0.

So substituting into the numerator: |2*6 + (-1)*0 + (-2)|. Let me compute that. 2*6 is 12, (-1)*0 is 0, and then -2. So 12 + 0 - 2 = 10. So absolute value of 10 is 10. Then the denominator is sqrt(2² + (-1)^2) which is sqrt(4 + 1) = sqrt(5). Therefore, the distance is 10 / sqrt(5). But they want it in simplest radical form. So rationalizing the denominator: 10 / sqrt(5) is equal to (10*sqrt(5)) / 5, which simplifies to 2*sqrt(5). So is that the answer? Wait, let me check again.

Alternatively, maybe I made a mistake in the sign when rearranging the equation. Let me check again. Original equation: y = 2x - 2. If I bring all terms to the left side: y - 2x + 2 = 0. Which is equivalent to -2x + y + 2 = 0. So in this case, A is -2, B is 1, C is 2. Then plugging into the formula: |-2*6 + 1*0 + 2| / sqrt((-2)^2 + 1^2). Let me compute that. Numerator: -12 + 0 + 2 = -10, absolute value is 10. Denominator: sqrt(4 + 1) = sqrt(5). So same result, 10 / sqrt(5) = 2*sqrt(5). So same answer. So regardless of which form I use, I get the same result. So that seems good.

But just to be thorough, maybe I can try another method to confirm. For example, using calculus or parametric equations. Let me think. The shortest distance from a point to a line is along the perpendicular. So maybe I can find the equation of the line perpendicular to y = 2x - 2 that passes through (6, 0), find the intersection point with the original line, and then compute the distance between (6, 0) and that intersection point.

Okay, let's try that. The original line has slope 2, so the perpendicular line will have slope -1/2. So the equation of the perpendicular line passing through (6, 0) is y - 0 = -1/2 (x - 6). Simplifying: y = (-1/2)x + 3. Now, find the intersection point between y = 2x - 2 and y = (-1/2)x + 3.

Set them equal: 2x - 2 = (-1/2)x + 3. Let me solve for x. Multiply both sides by 2 to eliminate the fraction: 4x - 4 = -x + 6. Then add x to both sides: 5x - 4 = 6. Add 4 to both sides: 5x = 10. Therefore, x = 2. Then plug back into one of the equations to find y. Let's use y = 2x - 2. So y = 2*2 - 2 = 4 - 2 = 2. Therefore, the intersection point is (2, 2). Now, compute the distance between (6, 0) and (2, 2).

Using the distance formula: sqrt[(6 - 2)^2 + (0 - 2)^2] = sqrt[(4)^2 + (-2)^2] = sqrt[16 + 4] = sqrt[20]. Simplify sqrt[20] = 2*sqrt(5). So same answer as before. So that confirms it. Therefore, the shortest distance is 2*sqrt(5).

Alternatively, maybe I can use vectors or parametric equations? Let me try parametric equations. Let me parametrize the line y = 2x - 2. Let me let x = t, so y = 2t - 2. So any point on the line can be written as (t, 2t - 2). Then the distance squared from (6, 0) to (t, 2t - 2) is (t - 6)^2 + (2t - 2 - 0)^2. Let me compute that: (t - 6)^2 + (2t - 2)^2. Expand both terms:

First term: (t - 6)^2 = t² - 12t + 36.

Second term: (2t - 2)^2 = 4t² - 8t + 4.

Adding them together: t² -12t + 36 + 4t² -8t +4 = 5t² -20t + 40.

To find the minimum distance, take derivative with respect to t and set to zero. But since this is a quadratic, the minimum occurs at vertex. The vertex of a quadratic at² + bt + c is at t = -b/(2a). Here, a = 5, b = -20. So t = -(-20)/(2*5) = 20/10 = 2. So t = 2. Then the point on the line is (2, 2*2 - 2) = (2, 2), same as before. Then distance squared is 5*(2)^2 -20*(2) +40 = 5*4 -40 +40 = 20. Therefore, distance is sqrt(20) = 2*sqrt(5). Same answer again. So that's consistent.

So all methods lead to the same answer. Therefore, I can be confident that the shortest distance is 2√5.

But just to recap, the formula method was straightforward. Let me check once again the formula. The formula is |Ax + By + C| / sqrt(A² + B²). So for the line y = 2x - 2, rearranged as 2x - y - 2 = 0. So A = 2, B = -1, C = -2. Then plugging in (6, 0):

|2*6 + (-1)*0 -2| / sqrt(2² + (-1)^2) = |12 - 0 -2| / sqrt(4 + 1) = |10| / sqrt(5) = 10 / sqrt(5) = 2*sqrt(5). Yep, same result.

Alternatively, if I had used the other form of the line equation, -2x + y + 2 = 0, then A = -2, B = 1, C = 2. Then plugging in (6, 0):

|(-2)*6 + 1*0 + 2| / sqrt((-2)^2 + 1^2) = |-12 + 0 + 2| / sqrt(4 + 1) = |-10| / sqrt(5) = 10 / sqrt(5) = 2*sqrt(5). Same answer.

So regardless of how I approach it, the answer is consistent. Therefore, I think that's solid.

Another way to think about it: maybe using coordinates and projection. The distance from the point to the line can be thought of as the length of the projection of the vector from any point on the line to the given point onto the direction perpendicular to the line. But that might be more complicated. But let me try.

Take a point on the line, say when x = 0, y = -2. So point (0, -2). The vector from (0, -2) to (6, 0) is (6 - 0, 0 - (-2)) = (6, 2). The direction vector of the line y = 2x - 2 is (1, 2), since the slope is 2. A perpendicular direction would be (-2, 1) or (2, -1). Wait, the direction vector of the line is (1, 2), so a perpendicular vector would have a dot product of zero. So (a, b) • (1, 2) = a + 2b = 0. So for example, (-2, 1) is perpendicular. So the distance is the absolute value of the projection of the vector (6, 2) onto the perpendicular direction.

Wait, the formula for the distance is |(vector) • perpendicular unit vector|. Let me recall. The distance is the length of the component of the vector from the point on the line to the given point in the direction perpendicular to the line. So if I have vector v from point P on the line to point Q (the given point), then the distance is |v ⋅ n| / |n|, where n is a normal vector to the line.

Alternatively, since the normal vector to the line y = 2x - 2 can be taken as (2, -1) or (-2, 1). Let me use (2, -1). Then the vector from (0, -2) to (6, 0) is (6, 2). The dot product of (6, 2) and (2, -1) is 6*2 + 2*(-1) = 12 - 2 = 10. The magnitude of the normal vector is sqrt(2² + (-1)^2) = sqrt(5). Therefore, the distance is |10| / sqrt(5) = 10 / sqrt(5) = 2*sqrt(5). Same answer again.

So that's another method, using vectors and dot product. So all methods converge to the same answer. Therefore, I can be confident that the shortest distance is indeed 2√5.

Just to recap, the key idea is that the shortest distance from a point to a line is the length of the perpendicular segment from the point to the line. There are multiple ways to calculate this: using the formula, using calculus to minimize the distance function, using parametric equations, using vector projections, or even geometrically by finding the intersection of the perpendicular line. All methods lead to the same result, which is reassuring.

Therefore, the answer is 2√5.

**Final Answer**
The shortest distance from the point $(6, 0)$ to the line $y = 2x - 2$ is \boxed{2\sqrt{5}}.

To find the **shortest distance** from a point to a line, we use the fact that the shortest distance is the **perpendicular distance** from the point to the line. This can be efficiently calculated using the **point-to-line distance formula**:

$$
\text{Distance} = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}
$$

---

### Step 1: Convert the line equation to standard form

We are given the line:

$$
y = 2x - 2
$$

Rewriting this in standard form $Ax + By + C = 0$:

$$
2x - y - 2 = 0
$$

So, the coefficients are:
- $A = 2$
- $B = -1$
- $C = -2$

---

### Step 2: Plug in the point coordinates

We are given the point $(6, 0)$. Let’s denote this as $(x_0, y_0) = (6, 0)$.

Substitute into the formula:

$$
\text{Distance} = \frac{|2(6) + (-1)(0) - 2|}{\sqrt{2^2 + (-1)^2}} = \frac{|12 - 0 - 2|}{\sqrt{4 + 1}} = \frac{10}{\sqrt{5}}
$$

---

### Step 3: Simplify the expression

To express the result in **simplest radical form**, we rationalize the denominator:

$$
\frac{10}{\sqrt{5}} = \frac{10\sqrt{5}}{5} = 2\sqrt{5}
$$

---

### Final Answer

$$
\boxed{2\sqrt{5}}
$$

[Assistant]
Explanation:
Okay, so I need to find the shortest distance from the point (6, 0) to the line y = 2x - 2. Hmm, let me think. I remember that the shortest distance from a point to a line is the perpendicular distance. So maybe I can use the formula for the distance from a point to a line? Wait, what's that formula again?

I think it's something like |Ax + By + C| divided by the square root of A squared plus B squared. But let me make sure. The general formula for the distance from a point (x₀, y₀) to the line Ax + By + C = 0 is |Ax₀ + By₀ + C| / sqrt(A² + B²). Right, that sounds familiar. So first, I need to write the given line equation in the standard form Ax + By + C = 0.

The given line is y = 2x - 2. Let me rearrange that. Subtract 2x from both sides: -2x + y + 2 = 0. Or maybe I can write it as 2x - y - 2 = 0. Wait, let me check. If I move all terms to the left side: y - 2x + 2 = 0? Wait, no. Let me do it step by step.

Original equation: y = 2x - 2. Subtract 2x from both sides: -2x + y = -2. Then add 2 to both sides: -2x + y + 2 = 0. So that's the standard form. Alternatively, multiplying both sides by -1 to make the coefficient of x positive: 2x - y - 2 = 0. Either way, both forms are acceptable. Let me pick one. Let me go with 2x - y - 2 = 0. So A is 2, B is -1, and C is -2.

Wait, but in the formula, it's Ax + By + C = 0. So if I have 2x - y - 2 = 0, then A = 2, B = -1, C = -2. So then plugging into the distance formula: |A*x₀ + B*y₀ + C| / sqrt(A² + B²). Let me check that again. So the point is (6, 0). So x₀ is 6, y₀ is 0.

So substituting into the numerator: |2*6 + (-1)*0 + (-2)|. Let me compute that. 2*6 is 12, (-1)*0 is 0, and then -2. So 12 + 0 - 2 = 10. So absolute value of 10 is 10. Then the denominator is sqrt(2² + (-1)^2) which is sqrt(4 + 1) = sqrt(5). Therefore, the distance is 10 / sqrt(5). But they want it in simplest radical form. So rationalizing the denominator: 10 / sqrt(5) is equal to (10*sqrt(5)) / 5, which simplifies to 2*sqrt(5). So is that the answer? Wait, let me check again.

Alternatively, maybe I made a mistake in the sign when rearranging the equation. Let me check again. Original equation: y = 2x - 2. If I bring all terms to the left side: y - 2x + 2 = 0. Which is equivalent to -2x + y + 2 = 0. So in this case, A is -2, B is 1, C is 2. Then plugging into the formula: |-2*6 + 1*0 + 2| / sqrt((-2)^2 + 1^2). Let me compute that. Numerator: -12 + 0 + 2 = -10, absolute value is 10. Denominator: sqrt(4 + 1) = sqrt(5). So same result, 10 / sqrt(5) = 2*sqrt(5). So same answer. So regardless of which form I use, I get the same result. So that seems good.

But just to be thorough, maybe I can try another method to confirm. For example, using calculus or parametric equations. Let me think. The shortest distance from a point to a line is along the perpendicular. So maybe I can find the equation of the line perpendicular to y = 2x - 2 that passes through (6, 0), find the intersection point with the original line, and then compute the distance between (6, 0) and that intersection point.

Okay, let's try that. The original line has slope 2, so the perpendicular line will have slope -1/2. So the equation of the perpendicular line passing through (6, 0) is y - 0 = -1/2 (x - 6). Simplifying: y = (-1/2)x + 3. Now, find the intersection point between y = 2x - 2 and y = (-1/2)x + 3.

Set them equal: 2x - 2 = (-1/2)x + 3. Let me solve for x. Multiply both sides by 2 to eliminate the fraction: 4x - 4 = -x + 6. Then add x to both sides: 5x - 4 = 6. Add 4 to both sides: 5x = 10. Therefore, x = 2. Then plug back into one of the equations to find y. Let's use y = 2x - 2. So y = 2*2 - 2 = 4 - 2 = 2. Therefore, the intersection point is (2, 2). Now, compute the distance between (6, 0) and (2, 2).

Using the distance formula: sqrt[(6 - 2)^2 + (0 - 2)^2] = sqrt[(4)^2 + (-2)^2] = sqrt[16 + 4] = sqrt[20]. Simplify sqrt[20] = 2*sqrt(5). So same answer as before. So that confirms it. Therefore, the shortest distance is 2*sqrt(5).

Alternatively, maybe I can use vectors or parametric equations? Let me try parametric equations. Let me parametrize the line y = 2x - 2. Let me let x = t, so y = 2t - 2. So any point on the line can be written as (t, 2t - 2). Then the distance squared from (6, 0) to (t, 2t - 2) is (t - 6)^2 + (2t - 2 - 0)^2. Let me compute that: (t - 6)^2 + (2t - 2)^2. Expand both terms:

First term: (t - 6)^2 = t² - 12t + 36.

Second term: (2t - 2)^2 = 4t² - 8t + 4.

Adding them together: t² -12t + 36 + 4t² -8t +4 = 5t² -20t + 40.

To find the minimum distance, take derivative with respect to t and set to zero. But since this is a quadratic, the minimum occurs at vertex. The vertex of a quadratic at² + bt + c is at t = -b/(2a). Here, a = 5, b = -20. So t = -(-20)/(2*5) = 20/10 = 2. So t = 2. Then the point on the line is (2, 2*2 - 2) = (2, 2), same as before. Then distance squared is 5*(2)^2 -20*(2) +40 = 5*4 -40 +40 = 20. Therefore, distance is sqrt(20) = 2*sqrt(5). Same answer again. So that's consistent.

So all methods lead to the same answer. Therefore, I can be confident that the shortest distance is 2√5.

But just to recap, the formula method was straightforward. Let me check once again the formula. The formula is |Ax + By + C| / sqrt(A² + B²). So for the line y = 2x - 2, rearranged as 2x - y - 2 = 0. So A = 2, B = -1, C = -2. Then plugging in (6, 0):

|2*6 + (-1)*0 -2| / sqrt(2² + (-1)^2) = |12 - 0 -2| / sqrt(4 + 1) = |10| / sqrt(5) = 10 / sqrt(5) = 2*sqrt(5). Yep, same result.

Alternatively, if I had used the other form of the line equation, -2x + y + 2 = 0, then A = -2, B = 1, C = 2. Then plugging in (6, 0):

|(-2)*6 + 1*0 + 2| / sqrt((-2)^2 + 1^2) = |-12 + 0 + 2| / sqrt(4 + 1) = |-10| / sqrt(5) = 10 / sqrt(5) = 2*sqrt(5). Same answer.

So regardless of how I approach it, the answer is consistent. Therefore, I think that's solid.

Another way to think about it: maybe using coordinates and projection. The distance from the point to the line can be thought of as the length of the projection of the vector from any point on the line to the given point onto the direction perpendicular to the line. But that might be more complicated. But let me try.

Take a point on the line, say when x = 0, y = -2. So point (0, -2). The vector from (0, -2) to (6, 0) is (6 - 0, 0 - (-2)) = (6, 2). The direction vector of the line y = 2x - 2 is (1, 2), since the slope is 2. A perpendicular direction would be (-2, 1) or (2, -1). Wait, the direction vector of the line is (1, 2), so a perpendicular vector would have a dot product of zero. So (a, b) • (1, 2) = a + 2b = 0. So for example, (-2, 1) is perpendicular. So the distance is the absolute value of the projection of the vector (6, 2) onto the perpendicular direction.

Wait, the formula for the distance is |(vector) • perpendicular unit vector|. Let me recall. The distance is the length of the component of the vector from the point on the line to the given point in the direction perpendicular to the line. So if I have vector v from point P on the line to point Q (the given point), then the distance is |v ⋅ n| / |n|, where n is a normal vector to the line.

Alternatively, since the normal vector to the line y = 2x - 2 can be taken as (2, -1) or (-2, 1). Let me use (2, -1). Then the vector from (0, -2) to (6, 0) is (6, 2). The dot product of (6, 2) and (2, -1) is 6*2 + 2*(-1) = 12 - 2 = 10. The magnitude of the normal vector is sqrt(2² + (-1)^2) = sqrt(5). Therefore, the distance is |10| / sqrt(5) = 10 / sqrt(5) = 2*sqrt(5). Same answer again.

So that's another method, using vectors and dot product. So all methods converge to the same answer. Therefore, I can be confident that the shortest distance is indeed 2√5.

Just to recap, the key idea is that the shortest distance from a point to a line is the length of the perpendicular segment from the point to the line. There are multiple ways to calculate this: using the formula, using calculus to minimize the distance function, using parametric equations, using vector projections, or even geometrically by finding the intersection of the perpendicular line. All methods lead to the same result, which is reassuring.

Therefore, the answer is 2√5.

**Final Answer**
The shortest distance from the point $(6, 0)$ to the line $y = 2x - 2$ is \boxed{2\sqrt{5}}.

To find the **shortest distance** from a point to a line, we use the fact that the shortest distance is the **perpendicular distance** from the point to the line. This can be efficiently calculated using the **point-to-line distance formula**:

$$
\text{Distance} = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}
$$

---

### Step 1: Convert the line equation to standard form

We are given the line:

$$
y = 2x - 2
$$

Rewriting this in standard form $Ax + By + C = 0$:

$$
2x - y - 2 = 0
$$

So, the coefficients are:
- $A = 2$
- $B = -1$
- $C = -2$

---

### Step 2: Plug in the point coordinates

We are given the point $(6, 0)$. Let’s denote this as $(x_0, y_0) = (6, 0)$.

Substitute into the formula:

$$
\text{Distance} = \frac{|2(6) + (-1)(0) - 2|}{\sqrt{2^2 + (-1)^2}} = \frac{|12 - 0 - 2|}{\sqrt{4 + 1}} = \frac{10}{\sqrt{5}}
$$

---

### Step 3: Simplify the expression

To express the result in **simplest radical form**, we rationalize the denominator:

$$
\frac{10}{\sqrt{5}} = \frac{10\sqrt{5}}{5} = 2\sqrt{5}
$$

---

### Final Answer

$$
\boxed{2\sqrt{5}}
$$

Solution:
The shortest line from the point $(6,0)$ to the given line will be perpendicular to it.  A line perpendicular to $y=2x-2$ will have a slope of $-1/2$.  This will give it a form of $y=-\frac{1}{2}x+b$.  Substituting the point $(6,0)$ that we know must lie on this line, we find: $$0=-\frac{1}{2}\cdot 6 +b$$ $$3=b$$ The equation of the perpendicular line is $y=-\frac{1}{2}x+3$.  Now, we can solve for the point where the two lines intersect: $$-\frac{1}{2}x+3=2x-2$$ $$5=\frac{5}{2}x$$ $$x=2$$ Plugging into either line, we find the point of intersection is $(2,2)$.  The coordinate plane now looks like: [asy]
size(150);
draw((-.5,0)--(7,0));
draw((0,-3)--(0,5));
draw((-.5,-3)--(4,6),linewidth(.7));
draw((6,0)--(0,3),linewidth(.7));
label("$(6,0)$",(6,0),S);
label("$(2,2)$",(2.3,2.1),E);
dot((2,2));
dot((6,0));
[/asy] The distance from the point $(6,0)$ to this point is: $$\sqrt{(6-2)^2+(0-2)^2}=\sqrt{16+4}=\boxed{2\sqrt{5}}$$