Explanation:
Okay, so I need to figure out how many y-intercepts the graph of the parabola x = y² - 4y - 1 has. Hmm, let me recall what a y-intercept is. If I remember correctly, a y-intercept is a point where the graph crosses the y-axis. On the coordinate plane, the y-axis is where x = 0. So, to find the y-intercepts, I should set x to 0 and solve for y. That makes sense. Let me write that down.

So, starting with the equation of the parabola: x = y² - 4y - 1. To find the y-intercepts, set x = 0:

0 = y² - 4y - 1.

Now, I need to solve this quadratic equation for y. Let me write that equation again:

y² - 4y - 1 = 0.

Hmm, quadratic equations can be solved using the quadratic formula, right? The quadratic formula is y = [-b ± √(b² - 4ac)] / (2a). In this equation, a is 1, b is -4, and c is -1. Let me plug those values into the formula.

First, let me compute the discriminant, which is b² - 4ac. So:

Discriminant D = (-4)² - 4 * 1 * (-1) = 16 - (-4) = 16 + 4 = 20.

Wait, that's 20. Since the discriminant is positive, that means there are two real solutions. Therefore, there are two real y-values that satisfy the equation when x = 0. Therefore, the graph crosses the y-axis at two points. So, does that mean there are two y-intercepts?

But hold on, let me make sure I didn't make a mistake here. Let me check my discriminant calculation again. The equation is y² - 4y - 1 = 0. So, a = 1, b = -4, c = -1. Then discriminant D = b² - 4ac = (-4)^2 - 4*(1)*(-1) = 16 - (-4) = 16 + 4 = 20. Yes, that's correct. So discriminant is 20, which is positive, so two real roots. Therefore, two y-intercepts.

But wait, let me think again. The original equation is x = y² - 4y - 1. This is a parabola that opens to the right or left, right? Since it's solved for x in terms of y. Normally, parabolas that open up or down are in the form y = ax² + bx + c, but this one is x = ay² + by + c, so it's a horizontal parabola. So, it opens either to the right or left. The standard form for a horizontal parabola is (y - k)² = 4p(x - h), which opens to the right if p > 0 and to the left if p < 0.

But maybe I don't need to go into that. The key point is that when we set x = 0, we are looking for points where the graph crosses the y-axis. Since solving 0 = y² - 4y - 1 gives two real solutions, that would mean two points where the graph crosses the y-axis. Therefore, two y-intercepts.

But let me also try solving the quadratic equation to see what the actual y-values are. Using the quadratic formula:

y = [4 ± √(20)] / 2.

Simplify √20. √20 is 2√5. So:

y = [4 ± 2√5]/2 = 2 ± √5.

So, the solutions are y = 2 + √5 and y = 2 - √5. These are two distinct real numbers, so there are two y-intercepts. Therefore, the answer should be 2.

But wait, let me check if there's any possibility that maybe the parabola doesn't cross the y-axis at all or maybe only once? For example, if the discriminant was zero, there would be one y-intercept, and if it was negative, no real y-intercepts. But since discriminant is positive, two intercepts. So, that seems right.

Alternatively, maybe I can graph the equation or think about it in another way. Let me try completing the square for the equation x = y² - 4y - 1. Maybe that will help me visualize it better.

Starting with x = y² - 4y - 1. Let me complete the square for the y terms. The y terms are y² - 4y. To complete the square, take half of the coefficient of y, which is -4, so half is -2, square it, which is 4. So, add and subtract 4:

x = (y² - 4y + 4 - 4) - 1.

Which simplifies to:

x = (y - 2)² - 4 - 1 = (y - 2)² - 5.

So, the equation can be written as (y - 2)² = x + 5. This is a horizontal parabola that opens to the right, with vertex at (-5, 2). So, the vertex is at (-5, 2). So, the parabola opens towards the positive x-direction. Therefore, it's going to extend infinitely to the right. Now, to find the y-intercepts, we set x = 0, which gives (y - 2)² = 0 + 5 = 5. Therefore, (y - 2)² = 5. Taking square roots, y - 2 = ±√5, so y = 2 ± √5. Which is exactly what I found earlier. Therefore, two solutions. Therefore, two y-intercepts.

So, that seems consistent. Therefore, the answer is 2. Therefore, the graph of the parabola x = y² - 4y - 1 has two y-intercepts.

But just to make sure, let me think again. Is there a case where even though the equation gives two solutions, maybe the graph doesn't actually cross the y-axis? For example, if the equation was something like x = y² - 4y + 4, which would be x = (y - 2)^2, which is a parabola opening to the right with vertex at (0, 2). In this case, setting x = 0 would give (y - 2)^2 = 0, so y = 2. So, only one y-intercept. But in our case, the equation is x = (y - 2)^2 - 5, so when x = 0, we have (y - 2)^2 = 5, which gives two solutions. So, that's two intercepts. So, that's correct.

Alternatively, if the equation was x = y² - 4y + 5, then setting x = 0 would give y² - 4y + 5 = 0, discriminant would be 16 - 20 = -4, which is negative, so no real solutions. Therefore, no y-intercepts. But in our case, discriminant is positive, so two intercepts.

Therefore, I think my reasoning is correct. Therefore, the answer is 2. Therefore, the number of y-intercepts is 2.

**Final Answer**
The graph of the parabola has \boxed{2} y-intercepts.

To determine how many **y-intercepts** the graph of the parabola
$$
x = y^2 - 4y - 1
$$
has, we need to understand what a **y-intercept** is. A y-intercept is a point where the graph crosses the **y-axis**, which occurs when the **x-coordinate is 0**. So, to find the y-intercepts, we set $ x = 0 $ and solve for $ y $.

---

### Step 1: Set $ x = 0 $

Substitute $ x = 0 $ into the equation:

$$
0 = y^2 - 4y - 1
$$

This is a quadratic equation in $ y $:

$$
y^2 - 4y - 1 = 0
$$

---

### Step 2: Solve the Quadratic Equation

We can solve this using the **quadratic formula**:

$$
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

Here, $ a = 1 $, $ b = -4 $, and $ c = -1 $. Plug these into the formula:

$$
y = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-1)}}{2(1)} = \frac{4 \pm \sqrt{16 + 4}}{2} = \frac{4 \pm \sqrt{20}}{2}
$$

Simplify $ \sqrt{20} = 2\sqrt{5} $, so:

$$
y = \frac{4 \pm 2\sqrt{5}}{2} = 2 \pm \sqrt{5}
$$

This gives **two distinct real solutions**:

$$
y = 2 + \sqrt{5} \quad \text{and} \quad y = 2 - \sqrt{5}
$$

---

### Step 3: Interpret the Result

Each solution corresponds to a **point on the y-axis**, where $ x = 0 $ and $ y $ takes one of these two values. Therefore, the graph of the parabola intersects the y-axis at **two distinct points**.

---

### Step 4: Confirm with Completing the Square (Optional)

To further confirm, we can rewrite the original equation in vertex form:

$$
x = y^2 - 4y - 1
$$

Complete the square for the $ y $-terms:

$$
x = (y^2 - 4y + 4) - 4 - 1 = (y - 2)^2 - 5
$$

So, the equation becomes:

$$
x = (y - 2)^2 - 5
$$

Setting $ x = 0 $:

$$
0 = (y - 2)^2 - 5 \Rightarrow (y - 2)^2 = 5 \Rightarrow y - 2 = \pm \sqrt{5} \Rightarrow y = 2 \pm \sqrt{5}
$$

Again, we find **two real solutions**, confirming the result.

---

### Final Conclusion

The graph of the parabola $ x = y^2 - 4y - 1 $ has **two y-intercepts**.

$$
\boxed{2}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many y-intercepts the graph of the parabola x = y² - 4y - 1 has. Hmm, let me recall what a y-intercept is. If I remember correctly, a y-intercept is a point where the graph crosses the y-axis. On the coordinate plane, the y-axis is where x = 0. So, to find the y-intercepts, I should set x to 0 and solve for y. That makes sense. Let me write that down.

So, starting with the equation of the parabola: x = y² - 4y - 1. To find the y-intercepts, set x = 0:

0 = y² - 4y - 1.

Now, I need to solve this quadratic equation for y. Let me write that equation again:

y² - 4y - 1 = 0.

Hmm, quadratic equations can be solved using the quadratic formula, right? The quadratic formula is y = [-b ± √(b² - 4ac)] / (2a). In this equation, a is 1, b is -4, and c is -1. Let me plug those values into the formula.

First, let me compute the discriminant, which is b² - 4ac. So:

Discriminant D = (-4)² - 4 * 1 * (-1) = 16 - (-4) = 16 + 4 = 20.

Wait, that's 20. Since the discriminant is positive, that means there are two real solutions. Therefore, there are two real y-values that satisfy the equation when x = 0. Therefore, the graph crosses the y-axis at two points. So, does that mean there are two y-intercepts?

But hold on, let me make sure I didn't make a mistake here. Let me check my discriminant calculation again. The equation is y² - 4y - 1 = 0. So, a = 1, b = -4, c = -1. Then discriminant D = b² - 4ac = (-4)^2 - 4*(1)*(-1) = 16 - (-4) = 16 + 4 = 20. Yes, that's correct. So discriminant is 20, which is positive, so two real roots. Therefore, two y-intercepts.

But wait, let me think again. The original equation is x = y² - 4y - 1. This is a parabola that opens to the right or left, right? Since it's solved for x in terms of y. Normally, parabolas that open up or down are in the form y = ax² + bx + c, but this one is x = ay² + by + c, so it's a horizontal parabola. So, it opens either to the right or left. The standard form for a horizontal parabola is (y - k)² = 4p(x - h), which opens to the right if p > 0 and to the left if p < 0.

But maybe I don't need to go into that. The key point is that when we set x = 0, we are looking for points where the graph crosses the y-axis. Since solving 0 = y² - 4y - 1 gives two real solutions, that would mean two points where the graph crosses the y-axis. Therefore, two y-intercepts.

But let me also try solving the quadratic equation to see what the actual y-values are. Using the quadratic formula:

y = [4 ± √(20)] / 2.

Simplify √20. √20 is 2√5. So:

y = [4 ± 2√5]/2 = 2 ± √5.

So, the solutions are y = 2 + √5 and y = 2 - √5. These are two distinct real numbers, so there are two y-intercepts. Therefore, the answer should be 2.

But wait, let me check if there's any possibility that maybe the parabola doesn't cross the y-axis at all or maybe only once? For example, if the discriminant was zero, there would be one y-intercept, and if it was negative, no real y-intercepts. But since discriminant is positive, two intercepts. So, that seems right.

Alternatively, maybe I can graph the equation or think about it in another way. Let me try completing the square for the equation x = y² - 4y - 1. Maybe that will help me visualize it better.

Starting with x = y² - 4y - 1. Let me complete the square for the y terms. The y terms are y² - 4y. To complete the square, take half of the coefficient of y, which is -4, so half is -2, square it, which is 4. So, add and subtract 4:

x = (y² - 4y + 4 - 4) - 1.

Which simplifies to:

x = (y - 2)² - 4 - 1 = (y - 2)² - 5.

So, the equation can be written as (y - 2)² = x + 5. This is a horizontal parabola that opens to the right, with vertex at (-5, 2). So, the vertex is at (-5, 2). So, the parabola opens towards the positive x-direction. Therefore, it's going to extend infinitely to the right. Now, to find the y-intercepts, we set x = 0, which gives (y - 2)² = 0 + 5 = 5. Therefore, (y - 2)² = 5. Taking square roots, y - 2 = ±√5, so y = 2 ± √5. Which is exactly what I found earlier. Therefore, two solutions. Therefore, two y-intercepts.

So, that seems consistent. Therefore, the answer is 2. Therefore, the graph of the parabola x = y² - 4y - 1 has two y-intercepts.

But just to make sure, let me think again. Is there a case where even though the equation gives two solutions, maybe the graph doesn't actually cross the y-axis? For example, if the equation was something like x = y² - 4y + 4, which would be x = (y - 2)^2, which is a parabola opening to the right with vertex at (0, 2). In this case, setting x = 0 would give (y - 2)^2 = 0, so y = 2. So, only one y-intercept. But in our case, the equation is x = (y - 2)^2 - 5, so when x = 0, we have (y - 2)^2 = 5, which gives two solutions. So, that's two intercepts. So, that's correct.

Alternatively, if the equation was x = y² - 4y + 5, then setting x = 0 would give y² - 4y + 5 = 0, discriminant would be 16 - 20 = -4, which is negative, so no real solutions. Therefore, no y-intercepts. But in our case, discriminant is positive, so two intercepts.

Therefore, I think my reasoning is correct. Therefore, the answer is 2. Therefore, the number of y-intercepts is 2.

**Final Answer**
The graph of the parabola has \boxed{2} y-intercepts.

To determine how many **y-intercepts** the graph of the parabola
$$
x = y^2 - 4y - 1
$$
has, we need to understand what a **y-intercept** is. A y-intercept is a point where the graph crosses the **y-axis**, which occurs when the **x-coordinate is 0**. So, to find the y-intercepts, we set $ x = 0 $ and solve for $ y $.

---

### Step 1: Set $ x = 0 $

Substitute $ x = 0 $ into the equation:

$$
0 = y^2 - 4y - 1
$$

This is a quadratic equation in $ y $:

$$
y^2 - 4y - 1 = 0
$$

---

### Step 2: Solve the Quadratic Equation

We can solve this using the **quadratic formula**:

$$
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

Here, $ a = 1 $, $ b = -4 $, and $ c = -1 $. Plug these into the formula:

$$
y = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-1)}}{2(1)} = \frac{4 \pm \sqrt{16 + 4}}{2} = \frac{4 \pm \sqrt{20}}{2}
$$

Simplify $ \sqrt{20} = 2\sqrt{5} $, so:

$$
y = \frac{4 \pm 2\sqrt{5}}{2} = 2 \pm \sqrt{5}
$$

This gives **two distinct real solutions**:

$$
y = 2 + \sqrt{5} \quad \text{and} \quad y = 2 - \sqrt{5}
$$

---

### Step 3: Interpret the Result

Each solution corresponds to a **point on the y-axis**, where $ x = 0 $ and $ y $ takes one of these two values. Therefore, the graph of the parabola intersects the y-axis at **two distinct points**.

---

### Step 4: Confirm with Completing the Square (Optional)

To further confirm, we can rewrite the original equation in vertex form:

$$
x = y^2 - 4y - 1
$$

Complete the square for the $ y $-terms:

$$
x = (y^2 - 4y + 4) - 4 - 1 = (y - 2)^2 - 5
$$

So, the equation becomes:

$$
x = (y - 2)^2 - 5
$$

Setting $ x = 0 $:

$$
0 = (y - 2)^2 - 5 \Rightarrow (y - 2)^2 = 5 \Rightarrow y - 2 = \pm \sqrt{5} \Rightarrow y = 2 \pm \sqrt{5}
$$

Again, we find **two real solutions**, confirming the result.

---

### Final Conclusion

The graph of the parabola $ x = y^2 - 4y - 1 $ has **two y-intercepts**.

$$
\boxed{2}
$$

Solution:
A $y$-intercept is a point on the graph that lies on the $y$-axis, so $x = 0$.  Hence, the number $y$-intercepts corresponds to the number of real solutions of the quadratic equation $y^2 - 4y - 1 = 0$.  The discriminant of this quadratic equation is $(-4)^2 + 4 \cdot 1 \cdot (-1) = 20$, which is positive, so the quadratic has two distinct real roots.  Therefore, the number of $y$-intercepts is $\boxed{2}$.

[asy]
size(150);
real ticklen=3;
real tickspace=2;

real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool

useticks=false, bool complexplane=false, bool usegrid=true) {

import graph;

real i;

if(complexplane) {

label("$\textnormal{Re}$",(xright,0),SE);

label("$\textnormal{Im}$",(0,ytop),NW);

} else {

label("$x$",(xright+0.4,-0.5));

label("$y$",(-0.5,ytop+0.2));

}

ylimits(ybottom,ytop);

xlimits( xleft, xright);

real[] TicksArrx,TicksArry;

for(i=xleft+xstep; i<xright; i+=xstep) {

if(abs(i) >0.1) {

TicksArrx.push(i);

}

}

for(i=ybottom+ystep; i<ytop; i+=ystep) {

if(abs(i) >0.1) {

TicksArry.push(i);

}

}

if(usegrid) {

xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray

(0.22),extend=true),p=invisible);//,above=true);

yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true),

p=invisible);//,Arrows);

}

if(useticks) {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry ,

pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx ,

pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));

} else {

xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));

yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));

}
};
real lowerx, upperx, lowery, uppery;
real f(real x) {return x^2 - 4*x - 1;}
lowery = -1;
uppery = 5;
rr_cartesian_axes(-6,5,lowery,uppery);
draw(reflect((0,0),(1,1))*(graph(f,lowery,uppery,operator ..)), red);
dot((0,2 + sqrt(5)));
dot((0,2 - sqrt(5)));
[/asy]


Explanation:
Okay, so I need to convert the fraction 21 divided by (2 squared times 5 to the seventh power) into a terminating decimal. Hmm, let me think. First, I remember that a fraction will have a terminating decimal if the denominator, after simplifying the fraction to its lowest terms, only has 2 and/or 5 as prime factors. Let me check if that's the case here.

The denominator given is 2² multiplied by 5⁷. So, the prime factors are just 2 and 5. That means the fraction should indeed convert to a terminating decimal. Good. Now, how do I actually perform the division?

Well, one method I remember is to adjust the denominator to be a power of 10, since 10 is 2 times 5. If I can make the denominator 10^n for some integer n, then the decimal will terminate. Let me try that.

The denominator is 2² * 5⁷. To make this a power of 10, I need to have the same number of 2s and 5s. Since there are 2² and 5⁷, I can multiply numerator and denominator by 2^(5) to make the number of 2s equal to the number of 5s. Wait, let me think again. The denominator has 2² and 5⁷. To make it 10^7, I need 7 2s and 7 5s. Since there are already 2², I need to multiply by 2^(7-2) = 2^5. Similarly, if I multiply by 2^5, the denominator becomes 2^(2+5) * 5^7 = 2^7 * 5^7 = (2*5)^7 = 10^7. Yes, that works. So, multiplying numerator and denominator by 2^5.

So, let me compute that. The original fraction is 21 / (2² * 5⁷). Multiply numerator and denominator by 2^5:

Numerator becomes 21 * 2^5.

Denominator becomes 10^7.

So, let me calculate 2^5 first. 2^5 is 32. Therefore, numerator is 21 * 32. Let me compute that. 20*32 = 640, and 1*32 = 32, so total is 640 + 32 = 672. Therefore, the fraction becomes 672 / 10^7.

Now, 10^7 is 10,000,000. So, 672 divided by 10,000,000. To write this as a decimal, I can move the decimal point in 672 seven places to the left. Let me do that step by step.

Starting with 672.0000000.

Moving decimal seven places left:

First place: 67.2000000

Second: 6.7200000

Third: 0.6720000

Fourth: 0.0672000

Fifth: 0.0067200

Sixth: 0.0006720

Seventh: 0.0000672

So, the decimal is 0.0000672. Let me check if that's correct. Alternatively, since 672 / 10^7 is the same as 672 divided by 10 million. So, 10 million is 10,000,000. So, 672 divided by 10,000,000. Let me write that as 0.0000672. Yes, because 1/10^7 is 0.0000001, so multiplying by 672 would be 672 * 0.0000001 = 0.0000672. That seems right.

But let me verify by another method. Maybe converting the original fraction step by step. Let me try dividing 21 by (2² * 5⁷). Let me compute the denominator first. 2² is 4, and 5⁷ is... let me calculate 5^7. 5^1=5, 5^2=25, 5^3=125, 5^4=625, 5^5=3125, 5^6=15625, 5^7=78125. So denominator is 4 * 78125. Let me compute that. 4 * 78125. Well, 78125 * 4. Let me do 78125 * 2 = 156250, then times 2 again is 312500. So denominator is 312,500. Therefore, the original fraction is 21 / 312,500.

Now, let me perform the division 21 ÷ 312,500. Hmm. Let me think. Since 312,500 is equal to 3.125 x 10^5. So, 21 divided by 3.125 x 10^5. Let me compute 21 / 3.125 first, then adjust the decimal.

21 divided by 3.125. Let me convert 3.125 to a fraction. 3.125 is 25/8. Because 3.125 = 3 + 1/8 = 25/8. Therefore, 21 divided by (25/8) is 21 * (8/25) = (21*8)/25. Let me compute 21*8. 20*8=160, 1*8=8, so 168. Therefore, 168/25. Now, 168 divided by 25. Let me compute that. 25*6=150, so 168-150=18. So, 6.72. Therefore, 168/25 = 6.72. Therefore, going back, 21 / 3.125 = 6.72. Therefore, the original division is 6.72 divided by 10^5, which is 6.72 / 100,000. Which is 0.0000672. So same result as before. So that confirms it. Therefore, the decimal is 0.0000672.

Alternatively, maybe I can do the division step by step. Let me try that. Let me write 21 divided by 312,500. Since 21 is less than 312,500, the decimal will start with 0. Let me add decimal places and zeros. So, 21.000000...

Divide by 312,500. Let me think how many decimal places. Alternatively, since I know that 1/312,500 is equal to 0.0000032, because 1/3125 is 0.00032, so dividing by 100 more would be 0.000000032? Wait, maybe I need to think differently. Wait, 312,500 is 3.125 x 10^5, so 1/312,500 is 1/(3.125 x 10^5) = (1/3.125) x 10^-5. As before, 1/3.125 is 0.32, so 0.32 x 10^-5 = 0.0000032. Therefore, 21 x 0.0000032 = ?

Let me compute 21 * 0.0000032. 20 * 0.0000032 = 0.000064, and 1 * 0.0000032 = 0.0000032. Adding them together: 0.000064 + 0.0000032 = 0.0000672. Yep, same result. So that's another way to get there.

So, all methods lead to 0.0000672. Therefore, the decimal is 0.0000672. Let me check if there are any trailing zeros or if I need to adjust. Since the original fraction is 21/(2²*5⁷), and we converted it to 672/10^7, which is 0.0000672. Since 672 is 672.0, and we have seven decimal places, so 0.0000672. Let me count the decimal places: 0.0000672. Let's see:

First decimal place: 0

Second: 0

Third: 0

Fourth: 0

Fifth: 6

Sixth: 7

Seventh: 2

Yes, seven decimal places. So that's correct. Therefore, the terminating decimal is 0.0000672.

Alternatively, maybe I can use another approach. Let me think. Since the denominator is 2² * 5⁷, which is 4 * 78125 = 312,500 as before. So, 21 divided by 312,500. Let me perform the division step by step.

So, 21 divided by 312,500. Let me write this as 21.000000 divided by 312,500. Since 312,500 goes into 21 zero times. So, 0. Then, add decimal places. Let me think of it as 21.000000 divided by 312,500. Let me multiply numerator and denominator by 10 to make it 210 divided by 3,125,000. Still not helpful. Maybe multiply numerator and denominator by 2 to get 42 divided by 625,000. Hmm, 625,000 is 6.25 x 10^5. Maybe not helpful. Alternatively, maybe divide numerator and denominator by something. Wait, 21 and 312,500. Let me check if they have common factors. 21 is 3*7. 312,500 is 2² * 5⁷. So, no common factors. Therefore, the fraction is already in simplest terms. Therefore, the decimal is 0.0000672 as before.

Alternatively, maybe I can use the fact that dividing by 5^7 is the same as multiplying by 2^7 and then dividing by 10^7? Wait, let me think. Let me try to manipulate the original fraction. The original fraction is 21/(2² * 5⁷). Let me write this as 21 * 2^(-2) * 5^(-7). If I want to write this as something over 10^n, since 10 is 2*5. Let me think. Let me adjust the exponents. Let me write 2^(-2) * 5^(-7) as 2^(-2) * 5^(-2) * 5^(-5) = (2*5)^(-2) * 5^(-5) = 10^(-2) * 5^(-5). Therefore, the original fraction is 21 * 10^(-2) * 5^(-5). Which is 21 / 10^2 * 1 / 5^5. Wait, 1/5^5 is 1/3125. So, 21 / 100 * 1/3125. Which is 21 / (100 * 3125) = 21 / 312,500, which is the same as before. So, 21 divided by 312,500. Which again leads to 0.0000672. So, same result.

Alternatively, maybe I can compute 21 divided by 5^7 first, then divide by 2². Let me try that. Let me compute 21 divided by 5^7. 5^7 is 78125. So, 21 / 78125. Let me compute that. Let me do 21 divided by 78125. Let me write that as 0.000... something. Let me think. 78125 goes into 21 zero times. So, 0. Then, 78125 goes into 210 zero times. 0.0. 78125 goes into 2100 zero times. 0.00. 78125 goes into 21000 zero times. 0.000. 78125 goes into 210000 zero times. 0.0000. 78125 goes into 2100000 two times? Let me check. 78125 * 2 = 156250. 78125 * 2.7 = 78125 * 2 + 78125 * 0.7 = 156250 + 54687.5 = 210,937.5. Which is more than 210,000. So, 2 times is 156,250. Subtract that from 210,000: 210,000 - 156,250 = 53,750. Bring down a zero? Wait, but I'm trying to do decimal division. Maybe this is getting too complicated. Alternatively, since I know that 1/78125 is 0.0000128, because 1/5^7. Wait, 1/5 is 0.2, 1/25 is 0.04, 1/125 is 0.008, 1/625 is 0.0016, 1/3125 is 0.00032, 1/15625 is 0.000064, 1/78125 is 0.0000128. Therefore, 21 * 0.0000128. Let me compute that. 20 * 0.0000128 = 0.000256, and 1 * 0.0000128 = 0.0000128. Adding them together: 0.000256 + 0.0000128 = 0.0002688. Then, since we had divided by 5^7 and then need to divide by 2², which is 4. So, 0.0002688 divided by 4. Let me compute that. 0.0002688 / 4 = 0.0000672. There we go! So that's another way to get there. So, that's 0.0000672. So, same answer again. Therefore, regardless of the method, I get 0.0000672.

Therefore, I can be confident that the decimal is 0.0000672. Let me just check once more with another approach. Let me use the fact that multiplying numerator and denominator by 2^5 gives me 672 / 10^7. So, 672 divided by 10^7. Let me write 672 as 6.72 x 10^2. Therefore, 6.72 x 10^2 divided by 10^7 is 6.72 x 10^(2-7) = 6.72 x 10^-5 = 0.0000672. Yep, that's another way to see it. So, 6.72 times 10^-5 is 0.0000672. Perfect.

So, all methods converge to the same decimal. Therefore, I think that's the correct answer. Just to recap, the key steps were recognizing that the denominator is a product of 2s and 5s, so the decimal terminates, then converting the denominator to a power of 10 by multiplying numerator and denominator by the appropriate power of 2, then simplifying to get the decimal. Alternatively, breaking down the division into smaller steps or using known decimal equivalents of fractions. All roads lead to Rome here.

Therefore, the final answer is 0.0000672.

**Final Answer**
The terminating decimal is \boxed{0.0000672}.

To convert the fraction
$$
\frac{21}{2^2 \cdot 5^7}
$$
into a **terminating decimal**, we begin by analyzing the denominator.

---

### Step 1: Understand the Nature of the Denominator

The denominator is
$$
2^2 \cdot 5^7
$$
which is a product of only the prime factors 2 and 5. This is a key property: **any fraction with a denominator that only contains the prime factors 2 and/or 5 will have a terminating decimal expansion**.

---

### Step 2: Convert the Denominator to a Power of 10

To convert the denominator into a power of 10, we need to balance the number of 2s and 5s. The current denominator has:

- $2^2$
- $5^7$

To make it a power of 10, we need the same number of 2s and 5s. So, we multiply both numerator and denominator by $2^5$ to get:

$$
\frac{21 \cdot 2^5}{(2^2 \cdot 5^7) \cdot 2^5} = \frac{21 \cdot 32}{2^7 \cdot 5^7} = \frac{672}{10^7}
$$

---

### Step 3: Convert to Decimal

Now, the fraction is
$$
\frac{672}{10^7}
$$
which is equivalent to moving the decimal point in 672 **seven places to the left**:

$$
672 \div 10^7 = 0.0000672
$$

---

### Final Answer

$$
\boxed{0.0000672}
$$

[Assistant]
Explanation:
Okay, so I need to convert the fraction 21 divided by (2 squared times 5 to the seventh power) into a terminating decimal. Hmm, let me think. First, I remember that a fraction will have a terminating decimal if the denominator, after simplifying the fraction to its lowest terms, only has 2 and/or 5 as prime factors. Let me check if that's the case here.

The denominator given is 2² multiplied by 5⁷. So, the prime factors are just 2 and 5. That means the fraction should indeed convert to a terminating decimal. Good. Now, how do I actually perform the division?

Well, one method I remember is to adjust the denominator to be a power of 10, since 10 is 2 times 5. If I can make the denominator 10^n for some integer n, then the decimal will terminate. Let me try that.

The denominator is 2² * 5⁷. To make this a power of 10, I need to have the same number of 2s and 5s. Since there are 2² and 5⁷, I can multiply numerator and denominator by 2^(5) to make the number of 2s equal to the number of 5s. Wait, let me think again. The denominator has 2² and 5⁷. To make it 10^7, I need 7 2s and 7 5s. Since there are already 2², I need to multiply by 2^(7-2) = 2^5. Similarly, if I multiply by 2^5, the denominator becomes 2^(2+5) * 5^7 = 2^7 * 5^7 = (2*5)^7 = 10^7. Yes, that works. So, multiplying numerator and denominator by 2^5.

So, let me compute that. The original fraction is 21 / (2² * 5⁷). Multiply numerator and denominator by 2^5:

Numerator becomes 21 * 2^5.

Denominator becomes 10^7.

So, let me calculate 2^5 first. 2^5 is 32. Therefore, numerator is 21 * 32. Let me compute that. 20*32 = 640, and 1*32 = 32, so total is 640 + 32 = 672. Therefore, the fraction becomes 672 / 10^7.

Now, 10^7 is 10,000,000. So, 672 divided by 10,000,000. To write this as a decimal, I can move the decimal point in 672 seven places to the left. Let me do that step by step.

Starting with 672.0000000.

Moving decimal seven places left:

First place: 67.2000000

Second: 6.7200000

Third: 0.6720000

Fourth: 0.0672000

Fifth: 0.0067200

Sixth: 0.0006720

Seventh: 0.0000672

So, the decimal is 0.0000672. Let me check if that's correct. Alternatively, since 672 / 10^7 is the same as 672 divided by 10 million. So, 10 million is 10,000,000. So, 672 divided by 10,000,000. Let me write that as 0.0000672. Yes, because 1/10^7 is 0.0000001, so multiplying by 672 would be 672 * 0.0000001 = 0.0000672. That seems right.

But let me verify by another method. Maybe converting the original fraction step by step. Let me try dividing 21 by (2² * 5⁷). Let me compute the denominator first. 2² is 4, and 5⁷ is... let me calculate 5^7. 5^1=5, 5^2=25, 5^3=125, 5^4=625, 5^5=3125, 5^6=15625, 5^7=78125. So denominator is 4 * 78125. Let me compute that. 4 * 78125. Well, 78125 * 4. Let me do 78125 * 2 = 156250, then times 2 again is 312500. So denominator is 312,500. Therefore, the original fraction is 21 / 312,500.

Now, let me perform the division 21 ÷ 312,500. Hmm. Let me think. Since 312,500 is equal to 3.125 x 10^5. So, 21 divided by 3.125 x 10^5. Let me compute 21 / 3.125 first, then adjust the decimal.

21 divided by 3.125. Let me convert 3.125 to a fraction. 3.125 is 25/8. Because 3.125 = 3 + 1/8 = 25/8. Therefore, 21 divided by (25/8) is 21 * (8/25) = (21*8)/25. Let me compute 21*8. 20*8=160, 1*8=8, so 168. Therefore, 168/25. Now, 168 divided by 25. Let me compute that. 25*6=150, so 168-150=18. So, 6.72. Therefore, 168/25 = 6.72. Therefore, going back, 21 / 3.125 = 6.72. Therefore, the original division is 6.72 divided by 10^5, which is 6.72 / 100,000. Which is 0.0000672. So same result as before. So that confirms it. Therefore, the decimal is 0.0000672.

Alternatively, maybe I can do the division step by step. Let me try that. Let me write 21 divided by 312,500. Since 21 is less than 312,500, the decimal will start with 0. Let me add decimal places and zeros. So, 21.000000...

Divide by 312,500. Let me think how many decimal places. Alternatively, since I know that 1/312,500 is equal to 0.0000032, because 1/3125 is 0.00032, so dividing by 100 more would be 0.000000032? Wait, maybe I need to think differently. Wait, 312,500 is 3.125 x 10^5, so 1/312,500 is 1/(3.125 x 10^5) = (1/3.125) x 10^-5. As before, 1/3.125 is 0.32, so 0.32 x 10^-5 = 0.0000032. Therefore, 21 x 0.0000032 = ?

Let me compute 21 * 0.0000032. 20 * 0.0000032 = 0.000064, and 1 * 0.0000032 = 0.0000032. Adding them together: 0.000064 + 0.0000032 = 0.0000672. Yep, same result. So that's another way to get there.

So, all methods lead to 0.0000672. Therefore, the decimal is 0.0000672. Let me check if there are any trailing zeros or if I need to adjust. Since the original fraction is 21/(2²*5⁷), and we converted it to 672/10^7, which is 0.0000672. Since 672 is 672.0, and we have seven decimal places, so 0.0000672. Let me count the decimal places: 0.0000672. Let's see:

First decimal place: 0

Second: 0

Third: 0

Fourth: 0

Fifth: 6

Sixth: 7

Seventh: 2

Yes, seven decimal places. So that's correct. Therefore, the terminating decimal is 0.0000672.

Alternatively, maybe I can use another approach. Let me think. Since the denominator is 2² * 5⁷, which is 4 * 78125 = 312,500 as before. So, 21 divided by 312,500. Let me perform the division step by step.

So, 21 divided by 312,500. Let me write this as 21.000000 divided by 312,500. Since 312,500 goes into 21 zero times. So, 0. Then, add decimal places. Let me think of it as 21.000000 divided by 312,500. Let me multiply numerator and denominator by 10 to make it 210 divided by 3,125,000. Still not helpful. Maybe multiply numerator and denominator by 2 to get 42 divided by 625,000. Hmm, 625,000 is 6.25 x 10^5. Maybe not helpful. Alternatively, maybe divide numerator and denominator by something. Wait, 21 and 312,500. Let me check if they have common factors. 21 is 3*7. 312,500 is 2² * 5⁷. So, no common factors. Therefore, the fraction is already in simplest terms. Therefore, the decimal is 0.0000672 as before.

Alternatively, maybe I can use the fact that dividing by 5^7 is the same as multiplying by 2^7 and then dividing by 10^7? Wait, let me think. Let me try to manipulate the original fraction. The original fraction is 21/(2² * 5⁷). Let me write this as 21 * 2^(-2) * 5^(-7). If I want to write this as something over 10^n, since 10 is 2*5. Let me think. Let me adjust the exponents. Let me write 2^(-2) * 5^(-7) as 2^(-2) * 5^(-2) * 5^(-5) = (2*5)^(-2) * 5^(-5) = 10^(-2) * 5^(-5). Therefore, the original fraction is 21 * 10^(-2) * 5^(-5). Which is 21 / 10^2 * 1 / 5^5. Wait, 1/5^5 is 1/3125. So, 21 / 100 * 1/3125. Which is 21 / (100 * 3125) = 21 / 312,500, which is the same as before. So, 21 divided by 312,500. Which again leads to 0.0000672. So, same result.

Alternatively, maybe I can compute 21 divided by 5^7 first, then divide by 2². Let me try that. Let me compute 21 divided by 5^7. 5^7 is 78125. So, 21 / 78125. Let me compute that. Let me do 21 divided by 78125. Let me write that as 0.000... something. Let me think. 78125 goes into 21 zero times. So, 0. Then, 78125 goes into 210 zero times. 0.0. 78125 goes into 2100 zero times. 0.00. 78125 goes into 21000 zero times. 0.000. 78125 goes into 210000 zero times. 0.0000. 78125 goes into 2100000 two times? Let me check. 78125 * 2 = 156250. 78125 * 2.7 = 78125 * 2 + 78125 * 0.7 = 156250 + 54687.5 = 210,937.5. Which is more than 210,000. So, 2 times is 156,250. Subtract that from 210,000: 210,000 - 156,250 = 53,750. Bring down a zero? Wait, but I'm trying to do decimal division. Maybe this is getting too complicated. Alternatively, since I know that 1/78125 is 0.0000128, because 1/5^7. Wait, 1/5 is 0.2, 1/25 is 0.04, 1/125 is 0.008, 1/625 is 0.0016, 1/3125 is 0.00032, 1/15625 is 0.000064, 1/78125 is 0.0000128. Therefore, 21 * 0.0000128. Let me compute that. 20 * 0.0000128 = 0.000256, and 1 * 0.0000128 = 0.0000128. Adding them together: 0.000256 + 0.0000128 = 0.0002688. Then, since we had divided by 5^7 and then need to divide by 2², which is 4. So, 0.0002688 divided by 4. Let me compute that. 0.0002688 / 4 = 0.0000672. There we go! So that's another way to get there. So, that's 0.0000672. So, same answer again. Therefore, regardless of the method, I get 0.0000672.

Therefore, I can be confident that the decimal is 0.0000672. Let me just check once more with another approach. Let me use the fact that multiplying numerator and denominator by 2^5 gives me 672 / 10^7. So, 672 divided by 10^7. Let me write 672 as 6.72 x 10^2. Therefore, 6.72 x 10^2 divided by 10^7 is 6.72 x 10^(2-7) = 6.72 x 10^-5 = 0.0000672. Yep, that's another way to see it. So, 6.72 times 10^-5 is 0.0000672. Perfect.

So, all methods converge to the same decimal. Therefore, I think that's the correct answer. Just to recap, the key steps were recognizing that the denominator is a product of 2s and 5s, so the decimal terminates, then converting the denominator to a power of 10 by multiplying numerator and denominator by the appropriate power of 2, then simplifying to get the decimal. Alternatively, breaking down the division into smaller steps or using known decimal equivalents of fractions. All roads lead to Rome here.

Therefore, the final answer is 0.0000672.

**Final Answer**
The terminating decimal is \boxed{0.0000672}.

To convert the fraction
$$
\frac{21}{2^2 \cdot 5^7}
$$
into a **terminating decimal**, we begin by analyzing the denominator.

---

### Step 1: Understand the Nature of the Denominator

The denominator is
$$
2^2 \cdot 5^7
$$
which is a product of only the prime factors 2 and 5. This is a key property: **any fraction with a denominator that only contains the prime factors 2 and/or 5 will have a terminating decimal expansion**.

---

### Step 2: Convert the Denominator to a Power of 10

To convert the denominator into a power of 10, we need to balance the number of 2s and 5s. The current denominator has:

- $2^2$
- $5^7$

To make it a power of 10, we need the same number of 2s and 5s. So, we multiply both numerator and denominator by $2^5$ to get:

$$
\frac{21 \cdot 2^5}{(2^2 \cdot 5^7) \cdot 2^5} = \frac{21 \cdot 32}{2^7 \cdot 5^7} = \frac{672}{10^7}
$$

---

### Step 3: Convert to Decimal

Now, the fraction is
$$
\frac{672}{10^7}
$$
which is equivalent to moving the decimal point in 672 **seven places to the left**:

$$
672 \div 10^7 = 0.0000672
$$

---

### Final Answer

$$
\boxed{0.0000672}
$$

Solution: A terminating decimal can be written in the form $\frac{a}{10^b}$, where $a$ and $b$ are integers. So we try to get a denominator of the form $10^b$: $$\frac{21}{2^2\cdot5^7}\cdot\frac{2^5}{2^5}=\frac{21\cdot32}{10^7}=\frac{672}{10^7}=\boxed{.0000672}.$$