Explanation:
Okay, so I need to simplify the square root of 242. Hmm, let me think. Simplifying square roots usually involves factoring the number under the root into a product of a perfect square and another number. Then, you can take the square root of the perfect square out of the radical. Let me try to remember the exact process.

First, I should factor 242 into its prime factors. Let me start by dividing by the smallest prime numbers. 242 is an even number, so it's divisible by 2. Let me do that first. 242 divided by 2 is... 121. Okay, so 242 = 2 × 121. Now, 121 seems familiar. Isn't that a perfect square? Yes! 11 times 11 is 121. So, 121 is 11 squared. Therefore, breaking it down further, 242 is 2 × 11².

So, writing that out, √242 = √(2 × 11²). Now, the square root of a product is the product of the square roots, right? So that would be √2 × √11². And the square root of 11 squared is just 11. So that simplifies to 11√2. Let me check if that's correct. If I square 11√2, I get (11)² × (√2)² = 121 × 2 = 242. Yep, that matches the original number. So that seems right.

Wait, but let me double-check if there's a larger perfect square factor that I might have missed. Let me think. When simplifying square roots, I should check if there are any other perfect squares in the factors. For example, maybe I could have broken down 242 differently? Let me try another approach. Let's see, 242 divided by 2 is 121, which we already did. What if I tried dividing by other numbers? Let me check if 242 is divisible by 3. 2 + 4 + 2 = 8, and 8 isn't divisible by 3, so no. Divided by 5? Ends with a 2, so no. Divided by 7? Let me check: 7 times 34 is 238, and 242 - 238 is 4, so no. 11? Well, 11 times 22 is 242? Wait, 11 × 22 is 242? Let me calculate that. 11 × 20 is 220, plus 11 × 2 is 22, so 220 + 22 = 242. Yes! So 242 is also 11 × 22. But 22 can be broken down into 2 × 11. So, that gives 11 × 2 × 11, which is the same as 2 × 11². So, same as before. So that's consistent. So, the prime factors are 2 × 11 × 11. Therefore, the square root of that is 11√2.

Is there any other way to factor 242? Let's see. Maybe 242 divided by something else. Let me see. 242 divided by 11 is 22, as we saw. Divided by 121 is 2. So, I think we've exhausted the factors. So, there's no larger perfect square factor than 121. So, the simplified form is indeed 11√2.

Wait, but just to make sure, let me check if there's another way to break it down. For example, maybe 242 can be expressed as a multiple of another perfect square? Let me check. The perfect squares less than 242 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. So, let me check if 242 is divisible by any of these squares except 1 and 121. Let me check 121 first, which we already did. 242 divided by 121 is 2. Then, 196 is 14 squared. 242 divided by 196 is... 1.234... which isn't an integer. 169 is 13 squared. 242 divided by 169 is about 1.43, not integer. 144 is 12 squared. 242 divided by 144 is roughly 1.68, not integer. 121 we already did. 100 is 10 squared. 242 divided by 100 is 2.42, not integer. Similarly, 81 is 9 squared. 242 divided by 81 is about 2.98, not integer. So none of these other perfect squares divide into 242 evenly except 1 and 121. Therefore, 121 is the largest perfect square factor. Therefore, the simplified form is indeed 11√2.

Alternatively, maybe I can think of breaking the square root into parts. Let me try that. So √242. Let me see if I can write 242 as a product of two numbers where one is a perfect square. For example, 242 = 121 × 2, which we already did. So √(121 × 2) = √121 × √2 = 11√2. That's the same as before.

Alternatively, if I tried writing 242 as something else. Let me think. Suppose I didn't think of 121 first. Maybe I could split it as 2 × 121, but maybe I thought of splitting it as 2 × 11 × 11. Either way, same result.

Let me also check with another example. Suppose I have √(2 × 11²). Then, since 11² is a perfect square, I take that out. So √(11²) is 11, and then the √2 remains. So 11√2. So, that's the simplified radical form.

Another thing to check: is 2 a prime number? Yes, so there's no more factors to break down. Therefore, √2 cannot be simplified further. So, the simplified form is 11√2.

Therefore, after all these steps and checks, I think the simplified form of √242 is 11√2.

**Final Answer**
The simplified form of $\sqrt{242}$ is \boxed{11\sqrt{2}}.

To simplify the square root of 242, we begin by factoring the number into its prime components. Understanding that square roots can often be simplified by identifying and separating out perfect squares, we follow this step-by-step process.

---

### Step 1: Prime Factorization of 242

We start by dividing 242 by its smallest prime factor:

$$
242 \div 2 = 121
$$

Now, 121 is recognized as a perfect square:

$$
121 = 11^2
$$

So the complete factorization of 242 is:

$$
242 = 2 \times 11^2
$$

---

### Step 2: Use the Square Root Property

We now apply the property of square roots:

$$
\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}
$$

Using this, we express the square root of 242 as:

$$
\sqrt{242} = \sqrt{2 \times 11^2} = \sqrt{2} \times \sqrt{11^2}
$$

Since the square root of a square is the base:

$$
\sqrt{11^2} = 11
$$

Therefore:

$$
\sqrt{242} = 11 \times \sqrt{2}
$$

---

### Step 3: Final Simplified Form

The expression $11\sqrt{2}$ is the simplified form of $\sqrt{242}$. No further simplification is possible because 2 is a prime number, and it is not a perfect square.

---

### Final Answer

$$
\boxed{11\sqrt{2}}
$$

[Assistant]
Explanation:
Okay, so I need to simplify the square root of 242. Hmm, let me think. Simplifying square roots usually involves factoring the number under the root into a product of a perfect square and another number. Then, you can take the square root of the perfect square out of the radical. Let me try to remember the exact process.

First, I should factor 242 into its prime factors. Let me start by dividing by the smallest prime numbers. 242 is an even number, so it's divisible by 2. Let me do that first. 242 divided by 2 is... 121. Okay, so 242 = 2 × 121. Now, 121 seems familiar. Isn't that a perfect square? Yes! 11 times 11 is 121. So, 121 is 11 squared. Therefore, breaking it down further, 242 is 2 × 11².

So, writing that out, √242 = √(2 × 11²). Now, the square root of a product is the product of the square roots, right? So that would be √2 × √11². And the square root of 11 squared is just 11. So that simplifies to 11√2. Let me check if that's correct. If I square 11√2, I get (11)² × (√2)² = 121 × 2 = 242. Yep, that matches the original number. So that seems right.

Wait, but let me double-check if there's a larger perfect square factor that I might have missed. Let me think. When simplifying square roots, I should check if there are any other perfect squares in the factors. For example, maybe I could have broken down 242 differently? Let me try another approach. Let's see, 242 divided by 2 is 121, which we already did. What if I tried dividing by other numbers? Let me check if 242 is divisible by 3. 2 + 4 + 2 = 8, and 8 isn't divisible by 3, so no. Divided by 5? Ends with a 2, so no. Divided by 7? Let me check: 7 times 34 is 238, and 242 - 238 is 4, so no. 11? Well, 11 times 22 is 242? Wait, 11 × 22 is 242? Let me calculate that. 11 × 20 is 220, plus 11 × 2 is 22, so 220 + 22 = 242. Yes! So 242 is also 11 × 22. But 22 can be broken down into 2 × 11. So, that gives 11 × 2 × 11, which is the same as 2 × 11². So, same as before. So that's consistent. So, the prime factors are 2 × 11 × 11. Therefore, the square root of that is 11√2.

Is there any other way to factor 242? Let's see. Maybe 242 divided by something else. Let me see. 242 divided by 11 is 22, as we saw. Divided by 121 is 2. So, I think we've exhausted the factors. So, there's no larger perfect square factor than 121. So, the simplified form is indeed 11√2.

Wait, but just to make sure, let me check if there's another way to break it down. For example, maybe 242 can be expressed as a multiple of another perfect square? Let me check. The perfect squares less than 242 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. So, let me check if 242 is divisible by any of these squares except 1 and 121. Let me check 121 first, which we already did. 242 divided by 121 is 2. Then, 196 is 14 squared. 242 divided by 196 is... 1.234... which isn't an integer. 169 is 13 squared. 242 divided by 169 is about 1.43, not integer. 144 is 12 squared. 242 divided by 144 is roughly 1.68, not integer. 121 we already did. 100 is 10 squared. 242 divided by 100 is 2.42, not integer. Similarly, 81 is 9 squared. 242 divided by 81 is about 2.98, not integer. So none of these other perfect squares divide into 242 evenly except 1 and 121. Therefore, 121 is the largest perfect square factor. Therefore, the simplified form is indeed 11√2.

Alternatively, maybe I can think of breaking the square root into parts. Let me try that. So √242. Let me see if I can write 242 as a product of two numbers where one is a perfect square. For example, 242 = 121 × 2, which we already did. So √(121 × 2) = √121 × √2 = 11√2. That's the same as before.

Alternatively, if I tried writing 242 as something else. Let me think. Suppose I didn't think of 121 first. Maybe I could split it as 2 × 121, but maybe I thought of splitting it as 2 × 11 × 11. Either way, same result.

Let me also check with another example. Suppose I have √(2 × 11²). Then, since 11² is a perfect square, I take that out. So √(11²) is 11, and then the √2 remains. So 11√2. So, that's the simplified radical form.

Another thing to check: is 2 a prime number? Yes, so there's no more factors to break down. Therefore, √2 cannot be simplified further. So, the simplified form is 11√2.

Therefore, after all these steps and checks, I think the simplified form of √242 is 11√2.

**Final Answer**
The simplified form of $\sqrt{242}$ is \boxed{11\sqrt{2}}.

To simplify the square root of 242, we begin by factoring the number into its prime components. Understanding that square roots can often be simplified by identifying and separating out perfect squares, we follow this step-by-step process.

---

### Step 1: Prime Factorization of 242

We start by dividing 242 by its smallest prime factor:

$$
242 \div 2 = 121
$$

Now, 121 is recognized as a perfect square:

$$
121 = 11^2
$$

So the complete factorization of 242 is:

$$
242 = 2 \times 11^2
$$

---

### Step 2: Use the Square Root Property

We now apply the property of square roots:

$$
\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}
$$

Using this, we express the square root of 242 as:

$$
\sqrt{242} = \sqrt{2 \times 11^2} = \sqrt{2} \times \sqrt{11^2}
$$

Since the square root of a square is the base:

$$
\sqrt{11^2} = 11
$$

Therefore:

$$
\sqrt{242} = 11 \times \sqrt{2}
$$

---

### Step 3: Final Simplified Form

The expression $11\sqrt{2}$ is the simplified form of $\sqrt{242}$. No further simplification is possible because 2 is a prime number, and it is not a perfect square.

---

### Final Answer

$$
\boxed{11\sqrt{2}}
$$

Solution: Factor 242 as $11^2 \cdot 2$. Then $\sqrt{242} = \sqrt{11^2} \cdot \sqrt2 = \boxed{11\sqrt2}$.