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 because if the graph crosses the y-axis, then at that point, the x-coordinate is 0.

Alright, so let me write down the equation again: x = y² - 4y - 1. If I want to find the y-intercepts, I need to substitute x with 0. Let me do that.

So substituting x = 0, the equation becomes:

0 = y² - 4y - 1

Now, this is a quadratic equation in terms of y. To find the solutions for y, I can use the quadratic formula. The quadratic formula is y = [-b ± √(b² - 4ac)] / (2a), where a, b, and c are coefficients from the quadratic equation ay² + by + c = 0.

In this case, comparing 0 = y² - 4y - 1 with the standard form, a = 1, b = -4, and c = -1. Let me plug these into the quadratic formula.

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

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

Wait, so the discriminant is 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.

But hold on, let me make sure I didn't make a mistake in calculating the discriminant. Let me check again:

Original equation after substitution: y² - 4y - 1 = 0.

So, a = 1, b = -4, c = -1.

Discriminant D = b² - 4ac = (-4)^2 - 4*(1)*(-1) = 16 - (-4) = 16 + 4 = 20. Yes, that's correct.

So, since the discriminant is positive, there are two distinct real roots. Therefore, there are two y-intercepts.

But wait, let me think again. The original equation is a parabola. But usually, parabolas can have different orientations. In this case, the equation is solved for x in terms of y, which means it's a parabola that opens either to the right or left, rather than up or down.

But regardless of the orientation, the y-intercepts are still points where x = 0. So, solving for x = 0 should give me the y-values where the graph crosses the y-axis.

But just to be thorough, let me try solving the quadratic equation step by step.

Starting with y² - 4y - 1 = 0.

Using the quadratic formula:

y = [4 ± √(20)] / 2.

Simplify √20. √20 is equal to √(4*5) = 2√5.

So, substituting back:

y = [4 ± 2√5]/2.

Divide numerator and denominator by 2:

y = 2 ± √5.

Therefore, the solutions are y = 2 + √5 and y = 2 - √5.

These are two distinct real numbers, so there are two y-intercepts.

But let me check if there's any possibility that these solutions might not be valid or if there's some other consideration.

For example, sometimes when dealing with equations, especially quadratics, you might have extraneous solutions, but in this case, since we're just solving for y when x = 0, and the quadratic equation is straightforward, there shouldn't be any issues.

Alternatively, maybe I can graph the equation mentally. The equation x = y² - 4y - 1 can be rewritten in vertex form to better understand its graph.

To convert it to vertex form, which is x = a(y - k)^2 + h, where (h, k) is the vertex.

Starting with x = y² - 4y - 1.

Let me complete the square for the y terms.

Take the coefficient of y, which is -4, divide by 2 to get -2, square it to get 4.

So, add and subtract 4:

x = (y² - 4y + 4 - 4) - 1

Which simplifies to:

x = (y² - 4y + 4) - 4 - 1

x = (y - 2)^2 - 5

So, the equation in vertex form is x = (y - 2)^2 - 5.

This is a parabola that opens to the right because the coefficient of the squared term is positive. The vertex is at (h, k) = (-5, 2).

So, the vertex is at (-5, 2). Since it opens to the right, the parabola extends infinitely to the right and left. However, since it's a horizontal parabola, its axis of symmetry is horizontal, which is the line y = 2.

Now, the y-intercepts occur where x = 0, so we can think of this as the points (0, y) on the parabola.

But since the vertex is at (-5, 2), which is to the left of the y-axis (since x = -5), and the parabola opens to the right, it should cross the y-axis at two points.

Wait, but how can a horizontal parabola cross the y-axis twice? Let me think.

If the parabola opens to the right, then as y increases or decreases, x increases. The vertex is at (-5, 2), so the minimum x-value is -5. Therefore, the parabola extends from x = -5 to the right. Therefore, when x = 0, which is to the right of the vertex, there should be two points where the parabola crosses the y-axis.

So, that aligns with our previous result of two solutions.

Alternatively, if I think about the graph, since the parabola opens to the right, it's symmetric about the line y = 2. Therefore, if it crosses the y-axis at two points, they should be symmetric with respect to y = 2.

Our solutions were y = 2 + √5 and y = 2 - √5. Let me check if they are symmetric around y = 2.

The average of the two y-values is [ (2 + √5) + (2 - √5) ] / 2 = (4)/2 = 2. So yes, they are symmetric around y = 2, which makes sense for the axis of symmetry.

Therefore, this further confirms that there are two y-intercepts.

But just to be absolutely sure, let me plug in the y-values back into the original equation to check if they result in x = 0.

First, take y = 2 + √5.

Compute x = (2 + √5)^2 - 4*(2 + √5) - 1.

Let me expand this step by step.

First, expand (2 + √5)^2:

= 2² + 2*2*√5 + (√5)^2

= 4 + 4√5 + 5

= 9 + 4√5

Then, compute -4*(2 + √5):

= -8 - 4√5

Now, combine all terms:

x = (9 + 4√5) + (-8 - 4√5) - 1

Simplify term by term:

9 - 8 - 1 + 4√5 - 4√5

Which is (0) + (0) = 0.

Perfect, that gives x = 0.

Now check y = 2 - √5.

Compute x = (2 - √5)^2 - 4*(2 - √5) - 1.

Again, expand (2 - √5)^2:

= 2² - 2*2*√5 + (√5)^2

= 4 - 4√5 + 5

= 9 - 4√5

Then compute -4*(2 - √5):

= -8 + 4√5

Combine all terms:

x = (9 - 4√5) + (-8 + 4√5) - 1

Simplify term by term:

9 - 8 - 1 - 4√5 + 4√5

Which is (0) + (0) = 0.

Perfect, that also gives x = 0.

Therefore, both solutions are valid, and they correspond to points (0, 2 + √5) and (0, 2 - √5).

Therefore, there are two y-intercepts.

But just to make sure I didn't miss anything, let me consider if there's any possibility of a repeated root or something else.

The discriminant was 20, which is positive, so two distinct real roots. Therefore, two distinct y-intercepts.

Alternatively, if the discriminant was zero, there would be one y-intercept (a repeated root), and if it was negative, there would be no real y-intercepts.

In this case, discriminant is positive, so two intercepts.

Therefore, the answer should be 2.

But let me think again if there's another way someone might approach this problem and perhaps make a mistake.

For example, someone might confuse x-intercepts with y-intercepts. But the question specifically asks for y-intercepts, so that's not the case here.

Alternatively, someone might think that since it's a parabola, it can only have one intercept, but that's only true for vertical parabolas (opening up/down). For horizontal parabolas (opening left/right), they can have two y-intercepts, one y-intercept, or none, depending on the equation.

In this case, since the equation is x = y² - 4y - 1, which we found has two solutions when x = 0, so two y-intercepts.

Another possible mistake could be miscalculating the discriminant. Let me check once more:

Original equation after substitution: y² - 4y - 1 = 0.

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

So, no mistake there.

Alternatively, someone might think that since the equation is quadratic in y, there can be at most two solutions, which is true, and since discriminant is positive, exactly two solutions.

Therefore, all checks confirm that there are two y-intercepts.

Therefore, the answer 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 terms of $ y $:

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

---

### Step 2: Solve the Quadratic Equation

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

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

For the equation $ y^2 - 4y - 1 = 0 $, the coefficients are:

- $ a = 1 $
- $ b = -4 $
- $ c = -1 $

Now compute the **discriminant**:

$$
D = b^2 - 4ac = (-4)^2 - 4(1)(-1) = 16 + 4 = 20
$$

Since the discriminant is **positive**, the equation has **two distinct real solutions**.

---

### Step 3: Find the Solutions

Using the quadratic formula:

$$
y = \frac{4 \pm \sqrt{20}}{2}
$$

Simplify $ \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} $, so:

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

Thus, the two y-values are:

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

These correspond to the **y-intercepts** of the graph, which are the points:

$$
(0, 2 + \sqrt{5}) \quad \text{and} \quad (0, 2 - \sqrt{5})
$$

---

### Step 4: Interpret the Result

Since the equation is a **horizontal parabola** (solved for $ x $ in terms of $ y $), it opens to the right or left. The vertex form of the equation is:

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

This confirms the parabola opens to the right and has a vertex at $ (-5, 2) $. Since the vertex is to the left of the y-axis and the parabola opens to the right, it **must cross the y-axis at two points**, which matches our earlier result.

---

### Final Answer

$$
\boxed{2}
$$

[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 because if the graph crosses the y-axis, then at that point, the x-coordinate is 0.

Alright, so let me write down the equation again: x = y² - 4y - 1. If I want to find the y-intercepts, I need to substitute x with 0. Let me do that.

So substituting x = 0, the equation becomes:

0 = y² - 4y - 1

Now, this is a quadratic equation in terms of y. To find the solutions for y, I can use the quadratic formula. The quadratic formula is y = [-b ± √(b² - 4ac)] / (2a), where a, b, and c are coefficients from the quadratic equation ay² + by + c = 0.

In this case, comparing 0 = y² - 4y - 1 with the standard form, a = 1, b = -4, and c = -1. Let me plug these into the quadratic formula.

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

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

Wait, so the discriminant is 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.

But hold on, let me make sure I didn't make a mistake in calculating the discriminant. Let me check again:

Original equation after substitution: y² - 4y - 1 = 0.

So, a = 1, b = -4, c = -1.

Discriminant D = b² - 4ac = (-4)^2 - 4*(1)*(-1) = 16 - (-4) = 16 + 4 = 20. Yes, that's correct.

So, since the discriminant is positive, there are two distinct real roots. Therefore, there are two y-intercepts.

But wait, let me think again. The original equation is a parabola. But usually, parabolas can have different orientations. In this case, the equation is solved for x in terms of y, which means it's a parabola that opens either to the right or left, rather than up or down.

But regardless of the orientation, the y-intercepts are still points where x = 0. So, solving for x = 0 should give me the y-values where the graph crosses the y-axis.

But just to be thorough, let me try solving the quadratic equation step by step.

Starting with y² - 4y - 1 = 0.

Using the quadratic formula:

y = [4 ± √(20)] / 2.

Simplify √20. √20 is equal to √(4*5) = 2√5.

So, substituting back:

y = [4 ± 2√5]/2.

Divide numerator and denominator by 2:

y = 2 ± √5.

Therefore, the solutions are y = 2 + √5 and y = 2 - √5.

These are two distinct real numbers, so there are two y-intercepts.

But let me check if there's any possibility that these solutions might not be valid or if there's some other consideration.

For example, sometimes when dealing with equations, especially quadratics, you might have extraneous solutions, but in this case, since we're just solving for y when x = 0, and the quadratic equation is straightforward, there shouldn't be any issues.

Alternatively, maybe I can graph the equation mentally. The equation x = y² - 4y - 1 can be rewritten in vertex form to better understand its graph.

To convert it to vertex form, which is x = a(y - k)^2 + h, where (h, k) is the vertex.

Starting with x = y² - 4y - 1.

Let me complete the square for the y terms.

Take the coefficient of y, which is -4, divide by 2 to get -2, square it to get 4.

So, add and subtract 4:

x = (y² - 4y + 4 - 4) - 1

Which simplifies to:

x = (y² - 4y + 4) - 4 - 1

x = (y - 2)^2 - 5

So, the equation in vertex form is x = (y - 2)^2 - 5.

This is a parabola that opens to the right because the coefficient of the squared term is positive. The vertex is at (h, k) = (-5, 2).

So, the vertex is at (-5, 2). Since it opens to the right, the parabola extends infinitely to the right and left. However, since it's a horizontal parabola, its axis of symmetry is horizontal, which is the line y = 2.

Now, the y-intercepts occur where x = 0, so we can think of this as the points (0, y) on the parabola.

But since the vertex is at (-5, 2), which is to the left of the y-axis (since x = -5), and the parabola opens to the right, it should cross the y-axis at two points.

Wait, but how can a horizontal parabola cross the y-axis twice? Let me think.

If the parabola opens to the right, then as y increases or decreases, x increases. The vertex is at (-5, 2), so the minimum x-value is -5. Therefore, the parabola extends from x = -5 to the right. Therefore, when x = 0, which is to the right of the vertex, there should be two points where the parabola crosses the y-axis.

So, that aligns with our previous result of two solutions.

Alternatively, if I think about the graph, since the parabola opens to the right, it's symmetric about the line y = 2. Therefore, if it crosses the y-axis at two points, they should be symmetric with respect to y = 2.

Our solutions were y = 2 + √5 and y = 2 - √5. Let me check if they are symmetric around y = 2.

The average of the two y-values is [ (2 + √5) + (2 - √5) ] / 2 = (4)/2 = 2. So yes, they are symmetric around y = 2, which makes sense for the axis of symmetry.

Therefore, this further confirms that there are two y-intercepts.

But just to be absolutely sure, let me plug in the y-values back into the original equation to check if they result in x = 0.

First, take y = 2 + √5.

Compute x = (2 + √5)^2 - 4*(2 + √5) - 1.

Let me expand this step by step.

First, expand (2 + √5)^2:

= 2² + 2*2*√5 + (√5)^2

= 4 + 4√5 + 5

= 9 + 4√5

Then, compute -4*(2 + √5):

= -8 - 4√5

Now, combine all terms:

x = (9 + 4√5) + (-8 - 4√5) - 1

Simplify term by term:

9 - 8 - 1 + 4√5 - 4√5

Which is (0) + (0) = 0.

Perfect, that gives x = 0.

Now check y = 2 - √5.

Compute x = (2 - √5)^2 - 4*(2 - √5) - 1.

Again, expand (2 - √5)^2:

= 2² - 2*2*√5 + (√5)^2

= 4 - 4√5 + 5

= 9 - 4√5

Then compute -4*(2 - √5):

= -8 + 4√5

Combine all terms:

x = (9 - 4√5) + (-8 + 4√5) - 1

Simplify term by term:

9 - 8 - 1 - 4√5 + 4√5

Which is (0) + (0) = 0.

Perfect, that also gives x = 0.

Therefore, both solutions are valid, and they correspond to points (0, 2 + √5) and (0, 2 - √5).

Therefore, there are two y-intercepts.

But just to make sure I didn't miss anything, let me consider if there's any possibility of a repeated root or something else.

The discriminant was 20, which is positive, so two distinct real roots. Therefore, two distinct y-intercepts.

Alternatively, if the discriminant was zero, there would be one y-intercept (a repeated root), and if it was negative, there would be no real y-intercepts.

In this case, discriminant is positive, so two intercepts.

Therefore, the answer should be 2.

But let me think again if there's another way someone might approach this problem and perhaps make a mistake.

For example, someone might confuse x-intercepts with y-intercepts. But the question specifically asks for y-intercepts, so that's not the case here.

Alternatively, someone might think that since it's a parabola, it can only have one intercept, but that's only true for vertical parabolas (opening up/down). For horizontal parabolas (opening left/right), they can have two y-intercepts, one y-intercept, or none, depending on the equation.

In this case, since the equation is x = y² - 4y - 1, which we found has two solutions when x = 0, so two y-intercepts.

Another possible mistake could be miscalculating the discriminant. Let me check once more:

Original equation after substitution: y² - 4y - 1 = 0.

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

So, no mistake there.

Alternatively, someone might think that since the equation is quadratic in y, there can be at most two solutions, which is true, and since discriminant is positive, exactly two solutions.

Therefore, all checks confirm that there are two y-intercepts.

Therefore, the answer 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 terms of $ y $:

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

---

### Step 2: Solve the Quadratic Equation

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

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

For the equation $ y^2 - 4y - 1 = 0 $, the coefficients are:

- $ a = 1 $
- $ b = -4 $
- $ c = -1 $

Now compute the **discriminant**:

$$
D = b^2 - 4ac = (-4)^2 - 4(1)(-1) = 16 + 4 = 20
$$

Since the discriminant is **positive**, the equation has **two distinct real solutions**.

---

### Step 3: Find the Solutions

Using the quadratic formula:

$$
y = \frac{4 \pm \sqrt{20}}{2}
$$

Simplify $ \sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5} $, so:

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

Thus, the two y-values are:

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

These correspond to the **y-intercepts** of the graph, which are the points:

$$
(0, 2 + \sqrt{5}) \quad \text{and} \quad (0, 2 - \sqrt{5})
$$

---

### Step 4: Interpret the Result

Since the equation is a **horizontal parabola** (solved for $ x $ in terms of $ y $), it opens to the right or left. The vertex form of the equation is:

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

This confirms the parabola opens to the right and has a vertex at $ (-5, 2) $. Since the vertex is to the left of the y-axis and the parabola opens to the right, it **must cross the y-axis at two points**, which matches our earlier result.

---

### Final Answer

$$
\boxed{2}
$$

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, 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 compute 2^5 first. 2^5 is 32. Therefore, numerator is 21 * 32. Let me calculate 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.

Alternatively, maybe I can do the division step by step. Let me try another approach to verify.

Original fraction: 21 / (2² * 5⁷). Let me compute the denominator first. 2² is 4, and 5⁷ is 78125. So, denominator is 4 * 78125. Let me compute that. 4 * 78125. Well, 78125 * 4. Let me compute 78125 * 2 = 156250, then times 2 again is 312500. So denominator is 312,500. Therefore, the fraction is 21 / 312,500. Now, let me perform this division.

So, 21 divided by 312,500. Let me write this as 21 ÷ 312500. Since 312500 goes into 21 zero times. Let me add decimal places. Let me think of it as 21.000000 divided by 312500.

Alternatively, maybe I can simplify the fraction first. Let me check if 21 and 312500 have any common factors. 21 factors are 3 and 7. Let me check if 3 divides into 312500. The sum of digits of 312500 is 3 + 1 + 2 + 5 + 0 + 0 = 11, which is not divisible by 3. So 3 is not a factor. Similarly, 7? Let me check if 7 divides into 312500. Dividing 312500 by 7. Well, 7*44642 = 312,494. Then 312,500 - 312,494 = 6. So remainder 6. Therefore, 7 is not a factor. Therefore, the fraction is already in simplest terms. So, 21/312500 is the simplified fraction.

So, to convert this to decimal, let me perform the division. Let me write 21 divided by 312500. Let me note that 21 divided by 312500 is equal to (21 / 3125) / 100. Wait, because 312500 is 3125 * 100. So, if I divide 21 by 3125 first, then divide by 100. Let me try that.

First, compute 21 divided by 3125. Let me do that division. 3125 goes into 21 zero times. Add decimal: 21.000000...

3125 goes into 21000 how many times? Let me compute 3125 * 6 = 18750. 3125 * 7 = 21875, which is too big. So 6 times. 6 * 3125 = 18750. Subtract from 21000: 21000 - 18750 = 2250. Bring down next 0: 22500.

3125 goes into 22500 how many times? 3125 * 7 = 21875. Subtract from 22500: 22500 - 21875 = 625. Bring down next 0: 6250.

3125 goes into 6250 exactly 2 times. 2 * 3125 = 6250. Subtract, remainder 0. So, so far, we have 0.00672. Wait, let me track the decimal places.

Original division: 21 divided by 3125. Let me think of it as 21.000000 divided by 3125.

First, 3125 goes into 21000 (after adding decimal and zeros) 6 times, as above. So that's 0.006... Then after that, we had 2250, then 22500, which gave us 7, then 6250, which gave us 2. So putting that together, the decimal is 0.00672. Then, since we had divided by 100 earlier, we need to divide by 100 again? Wait, no. Wait, original was 21 divided by 312500 = (21 / 3125) / 100. So, if 21 / 3125 is 0.00672, then dividing by 100 would be 0.0000672. Which matches the previous result. So that's consistent. Therefore, the decimal is 0.0000672.

So, both methods give me the same result. Therefore, I think that is correct. Let me check with another approach. Maybe convert the original fraction step by step.

Original fraction: 21 / (2² * 5⁷). Let me write this as 21 / (2² * 5² * 5⁵) = 21 / ( (2*5)² * 5⁵ ) = 21 / (10² * 5⁵). Then, 10² is 100, and 5⁵ is 3125. So denominator is 100 * 3125 = 312500, which is same as before. So again, same fraction. So, same result.

Alternatively, maybe I can write the denominator as 2² * 5⁷ = 2² * 5² * 5⁵ = (2*5)² * 5⁵ = 10² * 5⁵. Then, to make the denominator a power of 10, I need to multiply numerator and denominator by 2⁵, as before. Which gives 21 * 2⁵ / 10⁷. Which is 672 / 10⁷ = 0.0000672. Yep, same thing.

Alternatively, maybe I can use the fact that dividing by 5⁷ is the same as multiplying by 2⁷, but I need to adjust accordingly. Wait, let me think. If I have 21 divided by (2² * 5⁷), maybe I can write this as 21 * 2^(-2) * 5^(-7). But since 5^(-7) is 1/5^7, and 2^(-2) is 1/2². Alternatively, maybe I can write this as 21 * 2^(-2) * 5^(-7) = 21 * 2^(-2) * 5^(-7) * (2^7 / 2^7) = 21 * 2^(5) * 5^(-7) / 2^7 * 5^7? Wait, maybe that's complicating.

Alternatively, since 5^(-7) is 2^7 / 10^7. Wait, 5^(-7) = 1 / 5^7 = 2^7 / (2^7 * 5^7) ) = 2^7 / 10^7. Therefore, 21 / (2² * 5⁷) = 21 * 2^(-2) * 5^(-7) = 21 * 2^(-2) * 2^7 / 10^7 = 21 * 2^(5) / 10^7. Which is again 21 * 32 / 10^7 = 672 / 10^7 = 0.0000672. Yep, same result.

So, all methods lead to the same decimal. Therefore, I think that is correct. Let me just check with a calculator to be sure. Wait, I don't have a calculator here, but maybe I can do a rough check. Let me think about the magnitude.

The denominator is 2² * 5⁷ = 4 * 78125 = 312,500. So 21 divided by 312,500. Let me approximate. 21 divided by 300,000 is approximately 0.00007. Since 312,500 is a bit more than 300,000, the actual value should be a bit less than 0.00007. Our result is 0.0000672, which is 0.0000672, which is indeed slightly less than 0.00007. So that seems reasonable.

Alternatively, 0.0000672 is 6.72 x 10^-5. Let me check if 6.72 x 10^-5 multiplied by 312,500 gives 21. Let me compute 6.72 x 10^-5 * 312,500. 6.72 * 312,500 * 10^-5. 312,500 * 10^-5 is 3.125. Therefore, 6.72 * 3.125. Let me compute that. 6 * 3.125 = 18.75, 0.72 * 3.125. Let me compute 0.7 * 3.125 = 2.1875, and 0.02 * 3.125 = 0.0625. So total is 2.1875 + 0.0625 = 2.25. Therefore, total is 18.75 + 2.25 = 21. Perfect! So that checks out. Therefore, 0.0000672 is indeed the correct decimal.

Therefore, after all these steps and checks, I can be confident that the decimal is 0.0000672.

**Final Answer**
The decimal representation of $\frac{21}{2^2 \cdot 5^7}$ is \boxed{0.0000672}.

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

---

### Step 1: Understand the Denominator

The denominator is:
$$
2^2 \cdot 5^7
$$

This is a product of powers of 2 and 5, which are the prime factors of 10. Since the denominator only contains the prime factors 2 and 5, the fraction will indeed convert to a **terminating decimal**.

---

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

To make the denominator 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 $10^n$, we need equal powers of 2 and 5. So we multiply both numerator and denominator by $2^5$ to make the total number of 2s equal to 7:
$$
\frac{21}{2^2 \cdot 5^7} \cdot \frac{2^5}{2^5} = \frac{21 \cdot 2^5}{2^7 \cdot 5^7} = \frac{21 \cdot 32}{10^7}
$$

---

### Step 3: Simplify the Numerator

Now compute the numerator:
$$
21 \cdot 32 = 672
$$

So the fraction becomes:
$$
\frac{672}{10^7}
$$

---

### Step 4: Convert to Decimal

Since the denominator is $10^7 = 10,000,000$, we can write the fraction as:
$$
\frac{672}{10,000,000} = 0.0000672
$$

This is a **terminating decimal** with 7 decimal places.

---

### 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, 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 compute 2^5 first. 2^5 is 32. Therefore, numerator is 21 * 32. Let me calculate 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.

Alternatively, maybe I can do the division step by step. Let me try another approach to verify.

Original fraction: 21 / (2² * 5⁷). Let me compute the denominator first. 2² is 4, and 5⁷ is 78125. So, denominator is 4 * 78125. Let me compute that. 4 * 78125. Well, 78125 * 4. Let me compute 78125 * 2 = 156250, then times 2 again is 312500. So denominator is 312,500. Therefore, the fraction is 21 / 312,500. Now, let me perform this division.

So, 21 divided by 312,500. Let me write this as 21 ÷ 312500. Since 312500 goes into 21 zero times. Let me add decimal places. Let me think of it as 21.000000 divided by 312500.

Alternatively, maybe I can simplify the fraction first. Let me check if 21 and 312500 have any common factors. 21 factors are 3 and 7. Let me check if 3 divides into 312500. The sum of digits of 312500 is 3 + 1 + 2 + 5 + 0 + 0 = 11, which is not divisible by 3. So 3 is not a factor. Similarly, 7? Let me check if 7 divides into 312500. Dividing 312500 by 7. Well, 7*44642 = 312,494. Then 312,500 - 312,494 = 6. So remainder 6. Therefore, 7 is not a factor. Therefore, the fraction is already in simplest terms. So, 21/312500 is the simplified fraction.

So, to convert this to decimal, let me perform the division. Let me write 21 divided by 312500. Let me note that 21 divided by 312500 is equal to (21 / 3125) / 100. Wait, because 312500 is 3125 * 100. So, if I divide 21 by 3125 first, then divide by 100. Let me try that.

First, compute 21 divided by 3125. Let me do that division. 3125 goes into 21 zero times. Add decimal: 21.000000...

3125 goes into 21000 how many times? Let me compute 3125 * 6 = 18750. 3125 * 7 = 21875, which is too big. So 6 times. 6 * 3125 = 18750. Subtract from 21000: 21000 - 18750 = 2250. Bring down next 0: 22500.

3125 goes into 22500 how many times? 3125 * 7 = 21875. Subtract from 22500: 22500 - 21875 = 625. Bring down next 0: 6250.

3125 goes into 6250 exactly 2 times. 2 * 3125 = 6250. Subtract, remainder 0. So, so far, we have 0.00672. Wait, let me track the decimal places.

Original division: 21 divided by 3125. Let me think of it as 21.000000 divided by 3125.

First, 3125 goes into 21000 (after adding decimal and zeros) 6 times, as above. So that's 0.006... Then after that, we had 2250, then 22500, which gave us 7, then 6250, which gave us 2. So putting that together, the decimal is 0.00672. Then, since we had divided by 100 earlier, we need to divide by 100 again? Wait, no. Wait, original was 21 divided by 312500 = (21 / 3125) / 100. So, if 21 / 3125 is 0.00672, then dividing by 100 would be 0.0000672. Which matches the previous result. So that's consistent. Therefore, the decimal is 0.0000672.

So, both methods give me the same result. Therefore, I think that is correct. Let me check with another approach. Maybe convert the original fraction step by step.

Original fraction: 21 / (2² * 5⁷). Let me write this as 21 / (2² * 5² * 5⁵) = 21 / ( (2*5)² * 5⁵ ) = 21 / (10² * 5⁵). Then, 10² is 100, and 5⁵ is 3125. So denominator is 100 * 3125 = 312500, which is same as before. So again, same fraction. So, same result.

Alternatively, maybe I can write the denominator as 2² * 5⁷ = 2² * 5² * 5⁵ = (2*5)² * 5⁵ = 10² * 5⁵. Then, to make the denominator a power of 10, I need to multiply numerator and denominator by 2⁵, as before. Which gives 21 * 2⁵ / 10⁷. Which is 672 / 10⁷ = 0.0000672. Yep, same thing.

Alternatively, maybe I can use the fact that dividing by 5⁷ is the same as multiplying by 2⁷, but I need to adjust accordingly. Wait, let me think. If I have 21 divided by (2² * 5⁷), maybe I can write this as 21 * 2^(-2) * 5^(-7). But since 5^(-7) is 1/5^7, and 2^(-2) is 1/2². Alternatively, maybe I can write this as 21 * 2^(-2) * 5^(-7) = 21 * 2^(-2) * 5^(-7) * (2^7 / 2^7) = 21 * 2^(5) * 5^(-7) / 2^7 * 5^7? Wait, maybe that's complicating.

Alternatively, since 5^(-7) is 2^7 / 10^7. Wait, 5^(-7) = 1 / 5^7 = 2^7 / (2^7 * 5^7) ) = 2^7 / 10^7. Therefore, 21 / (2² * 5⁷) = 21 * 2^(-2) * 5^(-7) = 21 * 2^(-2) * 2^7 / 10^7 = 21 * 2^(5) / 10^7. Which is again 21 * 32 / 10^7 = 672 / 10^7 = 0.0000672. Yep, same result.

So, all methods lead to the same decimal. Therefore, I think that is correct. Let me just check with a calculator to be sure. Wait, I don't have a calculator here, but maybe I can do a rough check. Let me think about the magnitude.

The denominator is 2² * 5⁷ = 4 * 78125 = 312,500. So 21 divided by 312,500. Let me approximate. 21 divided by 300,000 is approximately 0.00007. Since 312,500 is a bit more than 300,000, the actual value should be a bit less than 0.00007. Our result is 0.0000672, which is 0.0000672, which is indeed slightly less than 0.00007. So that seems reasonable.

Alternatively, 0.0000672 is 6.72 x 10^-5. Let me check if 6.72 x 10^-5 multiplied by 312,500 gives 21. Let me compute 6.72 x 10^-5 * 312,500. 6.72 * 312,500 * 10^-5. 312,500 * 10^-5 is 3.125. Therefore, 6.72 * 3.125. Let me compute that. 6 * 3.125 = 18.75, 0.72 * 3.125. Let me compute 0.7 * 3.125 = 2.1875, and 0.02 * 3.125 = 0.0625. So total is 2.1875 + 0.0625 = 2.25. Therefore, total is 18.75 + 2.25 = 21. Perfect! So that checks out. Therefore, 0.0000672 is indeed the correct decimal.

Therefore, after all these steps and checks, I can be confident that the decimal is 0.0000672.

**Final Answer**
The decimal representation of $\frac{21}{2^2 \cdot 5^7}$ is \boxed{0.0000672}.

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

---

### Step 1: Understand the Denominator

The denominator is:
$$
2^2 \cdot 5^7
$$

This is a product of powers of 2 and 5, which are the prime factors of 10. Since the denominator only contains the prime factors 2 and 5, the fraction will indeed convert to a **terminating decimal**.

---

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

To make the denominator 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 $10^n$, we need equal powers of 2 and 5. So we multiply both numerator and denominator by $2^5$ to make the total number of 2s equal to 7:
$$
\frac{21}{2^2 \cdot 5^7} \cdot \frac{2^5}{2^5} = \frac{21 \cdot 2^5}{2^7 \cdot 5^7} = \frac{21 \cdot 32}{10^7}
$$

---

### Step 3: Simplify the Numerator

Now compute the numerator:
$$
21 \cdot 32 = 672
$$

So the fraction becomes:
$$
\frac{672}{10^7}
$$

---

### Step 4: Convert to Decimal

Since the denominator is $10^7 = 10,000,000$, we can write the fraction as:
$$
\frac{672}{10,000,000} = 0.0000672
$$

This is a **terminating decimal** with 7 decimal places.

---

### 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 $10^b$: $$\frac{21}{2^2\cdot5^7}\cdot\frac{2^5}{2^5}=\frac{21\cdot32}{10^7}=\frac{672}{10^7}=\boxed{.0000672}.$$