<Question>
Convert $\frac{21}{2^2 \cdot 5^7}$ to a terminating decimal.
</Question>

<think>
<Plan_1>
Okay, so I need to convert the fraction 21 divided by (2 squared times 5 to the 7th power) into a terminating decimal. Hmm, let me think about how to approach this. First, I remember that a fraction will convert to a terminating decimal if the denominator, after simplifying the fraction, has no prime factors other than 2 or 5. Since the denominator here is already in terms of 2s and 5s, that's a good start. But I need to figure out exactly how to convert this into a decimal.

Let me write down the fraction again to make sure I have it right: 21 over (2² * 5⁷). So that's 21 divided by (4 * 78125). Let me compute that denominator first.

<prompt_1.1> Simplify the expression 21 divided by (4 * 78125) as per the above plan. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
2 squared is 4, and 5 to the 7th power is... let's see, 5^1 is 5, 5^2 is 25, 5^3 is 125, 5^4 is 625, 5^5 is 3125, 5^6 is 15625, and 5^7 is 78125. So the denominator is 4 * 78125. Let me multiply those. 4 times 78125. Well, 4 times 70000 is 280000, and 4 times 8125 is 32500. So adding those together, 280000 + 32500 is 312500. So the denominator is 312500. Therefore, the fraction is 21/312500.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, to convert this fraction into a decimal, I can just perform the division, but that might be time-consuming. Alternatively, since the denominator is a product of powers of 2 and 5, I can adjust the fraction to have a denominator of 10^n, which would make it easy to convert to a decimal.

<prompt_2.1> Convert the fraction 21/312500 into a decimal by adjusting it to have a denominator of 10^n as per the above plan. </prompt_2.1>
-----

Alternatively, maybe another way to approach this is to separate the powers of 2 and 5. Let me try that. The original fraction is 21/(2²*5⁷). To convert this into a decimal, we need the denominator to be a power of 10, which is 2*5. So, we need to have the same number of 2s and 5s in the denominator. Currently, there are 2 twos and 7 fives. To make the denominator a power of 10, we need to multiply numerator and denominator by enough 2s or 5s to balance the exponents.

<prompt_2.2> Use an alternative method to convert the fraction 21/(2²*5⁷) into a decimal by balancing exponents as per the above plan. </prompt_2.2>
-----

Alternatively, if I had a different balance, but in this case, since we need to reach the higher exponent.
Since 5^7 is higher, we need to multiply by 2^5 to get both exponents to 7. That gives 10^7 as the denominator.

Alternatively, if there was a higher power of 2, we might have to multiply by 5s, but here 2s are fewer, so we multiply by 2s until the exponents match.

Another way to think about it is that the number of decimal places will be the maximum of the exponents in the denominator after balancing. Wait, no. Wait, when converting a fraction to a decimal, the number of decimal places is determined by the larger exponent between the numerator and denominator when expressed in terms of 10s. Wait, maybe not exactly. Let me recall.

<prompt_2.3> Consider alternative methods to confirm the conversion of the fraction to a decimal as per the above plan. </prompt_2.3>

Alternatively, if I had more 2s than 5s, then the number of decimal places would be determined by the number of 5s. Wait, let's think. Suppose the denominator is 2^a *5^b. Then, to make it 10^max(a,b), you need to multiply by 2^(max(a,b)-a) if a < b, or 5^(max(a,b)-b) if b < a. So the number of decimal places would be max(a,b). Wait, in this case, original denominator is 2^2 *5^7. So max(2,7)=7, so decimal has 7 places.

Therefore, 21/(2^2*5^7) can be written as (21*2^5)/(10^7) = (21*32)/10^7 = 672/10^7 = 0.000672.

Therefore, the answer is 0.000672.
-----

Alternatively, let's do the division step by step. 21 divided by 312500. Let's write this as 21 ÷ 312500. Let me express this division as a decimal by adding decimal places.

<prompt_2.4> Perform step-by-step division to confirm the conversion of the fraction to a decimal as per the above plan. </prompt_2.4>
</Plan_2>

<Execution_2>
<execution_2.1>
Let's see how to do that. The denominator is 2² * 5⁷. To make this a power of 10, I need the exponents of 2 and 5 to be equal. Since 10 is 2*5, each 10 in the denominator uses up one 2 and one 5. So, right now, there are 2 twos and 7 fives. The limiting factor here is the number of 2s. Since there are fewer 2s, we need to multiply numerator and denominator by enough 2s to make the number of 2s equal to the number of 5s. Wait, but 5s are more. So to make the denominator a power of 10, we need the exponents of 2 and 5 to be the same. So, since we have 2 twos and 7 fives, we need to multiply numerator and denominator by 2^(7-2) = 2^5 to make the number of 2s equal to 7. That way, the denominator becomes 2^7 * 5^7 = (2*5)^7 = 10^7. So, multiplying numerator and denominator by 2^5:

Numerator: 21 * 2^5
Denominator: (2^2 * 5^7) * 2^5 = 2^(2+5) * 5^7 = 2^7 * 5^7 = 10^7

So the fraction becomes (21 * 32) / 10^7. Let me compute 21 * 32. 20*32=640, and 1*32=32, so total is 640 +32=672. Therefore, 672 divided by 10^7.
</execution_2.1>

<execution_2.2>
Since there are more 5s, we need to multiply numerator and denominator by 2^(7-2) = 2^5 to get the denominator's 2s up to 7. That way, the denominator becomes 2^(2+5) *5^7 = 2^7 *5^7 = (2*5)^7 = 10^7. Then, the numerator becomes 21 * 2^5. Let me compute that.

2^5 is 32. So 21 * 32. Let's calculate that: 20*32=640, and 1*32=32, so total is 640 +32=672. So the numerator is 672, and the denominator is 10^7. Therefore, the fraction is 672/10,000,000. Which is 0.0000672. Wait, but 672 divided by 10^7. Let me confirm: 10^7 is 10,000,000. So 672 divided by 10,000,000. Moving the decimal point seven places to the left: 672 becomes 0.0000672. So that's 0.0000672.
</execution_2.2>

<execution_2.3>
For example, 1/8 is 0.125, which is 3 decimal places, and 8 is 2^3. So in that case, the number of decimal places is equal to the exponent in the denominator when it's a power of 2. Similarly, 1/5 is 0.2, which is one decimal place, matching the exponent of 5^1. So in general, if the denominator is 2^a *5^b, the number of decimal places is the maximum of a and b. Wait, in this case, original denominator is 2^2 *5^7. So max(2,7)=7, so 7 decimal places. Which matches our previous result.
</execution_2.3>

<execution_2.4>
Since 312500 is larger than 21, we know the result will be less than 1, so we start with 0. and then proceed to add zeros until we can divide.

But maybe it's easier to use the fact that multiplying numerator and denominator by 2^5 gives us 21*32 / 10^7, which is 672/10^7 = 0.000672.
</execution_2.4>
</Execution_2>

<Plan_3>
Based on execution_2.1:
But wait, 10^7 is 10,000,000. So 672 divided by 10,000,000. Let me write that as a decimal.
Dividing by 10^7 moves the decimal point seven places to the left. Starting with 672, which is 672.0, moving the decimal seven places left: 6.72 becomes 0.0672, then 0.00672, 0.000672, but wait, let me count properly.

<prompt_3.1> Convert the fraction 672/10,000,000 into a decimal by moving the decimal point as per the above plan. </prompt_3.1>
-----

Based on execution_2.2:
But wait, earlier I thought of multiplying by 2^5, but now I see that maybe I need to check if that's correct.

<prompt_3.2> Re-evaluate the alternative method and confirm the decimal conversion as per the above plan. </prompt_3.2>

But let me verify this.

Alternatively, if I consider the original denominator is 2^2 *5^7. To convert to a power of 10, we need to have the same number of 2s and 5s.

<prompt_3.3> Verify the alternative method and confirm the decimal conversion as per the above plan. </prompt_3.3>
-----

Based on execution_2.4:
Alternatively, let's do the division step by step. 21 ÷ 312500. Let's express this division as a decimal by adding decimal places.

<prompt_3.4> Confirm the decimal result using step-by-step division as per the above plan. </prompt_3.4>
</Plan_3>

<Execution_3>
<execution_3.1>
Starting with 672, moving the decimal one place left: 67.2, two places: 6.72, three places: 0.672, four places: 0.0672, five places: 0.00672, six places: 0.000672, seven places: 0.0000672. Wait, that seems off. Let me check again.

Wait, 10^7 is 10 million. So 1/10,000,000 is 0.0000001. Therefore, 672/10,000,000 is 672 * 0.0000001. Which is 0.0000672. Wait, so which is correct? Let me verify with exponents.

10^7 is 10,000,000. So 672 divided by 10^7 is 672 * 10^-7. So 672 * 10^-7. 672 is 6.72 * 10^2. So 6.72 * 10^2 * 10^-7 = 6.72 * 10^-5. Which is 0.0000672.
</execution_3.1>

<execution_3.2>
Wait, let's check again. Original denominator is 2^2*5^7. To make it 10^n, we need to have n equal to the maximum of 2 and 7, which is 7. So we need to multiply numerator and denominator by 2^(7-2) = 2^5. So denominator becomes 2^7*5^7, numerator becomes 21*2^5. So that's 21*32=672. So 672 / 10^7. 10^7 is 10,000,000. So 672 divided by 10,000,000. Let's write that as a decimal: 672 ÷ 10,000,000. Since 10,000,000 has 7 zeros, we move the decimal point in 672 seven places to the left. Starting from 672.0, moving the decimal one place: 67.20, two places: 6.720, three: 0.6720, four: 0.06720, five: 0.006720, six: 0.0006720, seven: 0.0000672. Wait, that's different from before. Wait, perhaps I made a mistake here. Let me check again.

Wait, 10^7 is 10,000,000. So 672 divided by 10,000,000. Let's write 672 as 672.0. Dividing by 10 moves the decimal one place left: 67.2, then another ten: 6.72, another ten: 0.672, another ten: 0.0672, another ten: 0.00672, another ten: 0.000672, another ten: 0.0000672. So yes, that's 6.72 times 10^-6, which is 0.00000672? Wait, no, wait. Wait, 10,000,000 is 10^7, so 672 / 10^7 is 672 * 10^-7. 672 * 10^-7 is 6.72 * 10^-5, which is 0.0000672. Wait, this is conflicting with my previous calculation. Let me check with exponents. 10^7 is 10,000,000. So 672 divided by 10,000,000. Let's write 672 as 6.72 * 10^2. Then, dividing by 10^7 gives 6.72 * 10^(2-7) = 6.72 * 10^-5. 10^-5 is 0.00001, so 6.72 * 0.00001 is 0.0000672. So that's correct. So 0.0000672. But wait, that's 6.72 times 10^-5, which is 0.0000672. But when I moved the decimal seven places, I thought it was 0.0000672, but actually, moving the decimal seven places left from 672.0 would be: 1st place: 67.2, 2nd: 6.72, 3rd: 0.672, 4th: 0.0672, 5th: 0.00672, 6th: 0.000672, 7th: 0.0000672. So yes, that's correct. So the decimal is 0.0000672.
</execution_3.2>

<execution_3.3>
Since there are 2 twos and 7 fives, we need to multiply numerator and denominator by 5^(7-2) =5^5 to make the number of 5s equal to 7. Wait, but that would make the denominator 2^2*5^7*5^5=2^2*5^12. That doesn't help. Wait, no, that approach is wrong. Wait, perhaps I need to make the exponents of 2 and 5 equal. So, if I have 2^2 and 5^7, I need to get them to the same exponent. To do that, I can either increase the number of 2s by multiplying by 2s or decrease the number of 5s by multiplying by 5s. Since 5s are more, perhaps it's better to multiply numerator and denominator by 2^(7-2)=2^5 to make the number of 2s equal to 7. Then the denominator becomes 2^(2+5)*5^7=2^7*5^7=10^7. Then numerator becomes 21*2^5=21*32=672. Therefore, 672/10^7=0.0000672. Wait, but that seems different from the previous result. Wait, no, 672 divided by 10^7 is 0.0000672? Wait, no. Wait, 10^7 is 10,000,000. So 672 divided by 10,000,000. Let's compute that.

672 ÷ 10,000,000. Since 672 ÷ 10 = 67.2, ÷10 again = 6.72, ÷10 = 0.672, ÷10 = 0.0672, ÷10 = 0.00672, ÷10 = 0.000672, ÷10 = 0.0000672. Wait, but that's 7 divisions by 10, which is equivalent to dividing by 10^7. So 672 ÷ 10^7 = 0.0000672. But that seems like 7 decimal places. Let me check: 10^1 is 10, 10^2 is 100, ...., 10^7 is 10,000,000. So 672 divided by 10,000,000. Let's write 672 as 672.0. Moving the decimal point seven places to the left: first division: 67.2, second: 6.72, third: 0.672, fourth: 0.0672, fifth: 0.00672, sixth: 0.000672, seventh: 0.0000672. So yes, that's correct. So the decimal is 0.0000672.
</execution_3.3>

<execution_3.4>
Since 312500 is larger than 21, we write 0. and then proceed. 21 ÷ 312500:

312500 goes into 21 zero times. Add a decimal point and a zero: 210 ÷ 312500, still less than 1. Add another zero: 2100 ÷ 312500, still less. Add another zero: 21000 ÷ 312500, still less. Add another zero: 210000 ÷ 312500. 312500 goes into 210000 zero times. Add another zero: 2100000 ÷ 312500. Let's compute how many times 312500 fits into 2100000. 312500 * 6 = 1,875,000. 312500 * 7 = 2,187,500 which is too much. So 6 times. 6*312500=1,875,000. Subtract that from 2,100,000: 2,100,000 - 1,875,000 = 225,000. Bring down the next zero: 2,250,000. 312500 * 7 = 2,187,500. Subtract: 2,250,000 - 2,187,500 = 62,500. Bring down a zero: 625,000. 312500 * 2 = 625,000. So that's 2 times. So putting it all together: 0.000672. So that's 0.000672.
</execution_3.4>
</Execution_3>

<Plan_4>
Based on execution_3.1:
But wait, that contradicts my initial thought. Let me check this again.
Wait, maybe I made a mistake in moving the decimal. Let me verify with a smaller number.

<prompt_4.1> Verify the decimal conversion by considering the smaller number example as per the above plan. </prompt_4.1>
-----

Based on execution_3.2, execution_3.3:
But when I did the division step by step, I got 0.0000672. Wait, but that's conflicting. Let me check again.

<prompt_4.2> Reconcile the decimal results from different methods as per the above plan. </prompt_4.2>
-----

Based on execution_3.4:
But let me confirm once more with the original fraction. 21 divided by 312500. Let me compute 21/312500.

<prompt_4.3> Confirm the final result using the original fraction and confirm the number of decimal places as per the above plan. </prompt_4.3>
</Plan_4>

<Execution_4>
<execution_4.1>
For example, 1/8 is 0.125. 8 is 2^3. So 1/2^3 is 0.125, which is three decimal places. Similarly, 1/5 is 0.2, which is one decimal place. So the number of decimal places seems to be the exponent in the denominator when it's a single prime. But when the denominator is a product of primes, like 2^2*5^7, the number of decimal places is the maximum of the exponents, which is 7.
</execution_4.1>

<execution_4.2>
For example, 1/10 is 0.1, 1/100 is 0.01, 1/1000 is 0.001, so 1/10^7 is 0.0000001. So 672 * 0.0000001 is 0.0000672. Therefore, 672/10,000,000 is 0.0000672. So that's correct. But when I moved the decimal seven places, I thought it was 0.0000672, but actually, 672 shifted seven places left is 0.0000672. So that's correct. Therefore, the decimal is 0.0000672.
</execution_4.2>

<execution_4.3>
Alternatively, 312500 is 3125*100, which is 5^5*100. So 5^5 is 3125, 100 is 10^2. So denominator is 5^5*10^2. Then the original fraction is 21/(5^5*10^2). If I multiply numerator and denominator by 2^5 to make the denominator 10^7, as before, giving 21*32/(5^5*2^5*10^2) = 672/(10^5*10^2) = 672/10^7. So yes, that's 0.0000672. But when I did the step-by-step division, I got 0.000672. Wait, that's a discrepancy. Which one is correct?

Wait, perhaps I made a mistake in the step-by-step division. Let me redo that. 21 ÷ 312500.

Start with 21 ÷ 312500. Since 312500 is 3.125 x 10^5, so 21 ÷ (3.125 x 10^5) = (21 ÷ 3.125) x 10^-5. Let's compute 21 ÷ 3.125. 3.125 is 5/1.6, but maybe easier to note that 3.125 x 6.72 = 21. Because 3.125 x 6 = 18.75, 3.125 x 0.72 = 2.25, so 18.75 + 2.25 = 21. So 3.125 x 6.72 = 21. Therefore, 21 ÷ 3.125 = 6.72. Therefore, 21 ÷ 3.125 x 10^-5 = 6.72 x 10^-5 = 0.0000672. So that's 0.0000672. But when I did the long division step by step, I ended up with 0.000672. That suggests I made an error in the long division method. Let me check that again.

Starting with 21 ÷ 312500:

- 312500 goes into 21 zero times. Add a decimal point and a zero: 210 ÷ 312500, still zero. Add another zero: 2100 ÷ 312500, zero. Add another zero: 21000 ÷ 312500, zero. Add another zero: 210000 ÷ 312500, zero. Add another zero: 2100000 ÷ 312500. How many times does 312500 go into 2100000? 312500 * 6 = 1,875,000. Subtract that from 2,100,000: 225,000. Bring down the next zero: 2,250,000. 312500 *7 = 2,187,500. Subtract: 62,500. Bring down a zero: 625,000. 312500 *2 = 625,000. So total is 6 (from first division), then 7, then 2. So the decimal is 0.000672. Wait, but according to the other method, it's 0.0000672. There's a discrepancy here. Which one is correct?

Wait, there's a mistake here. Let's compute 21/312500. Let's convert both numbers to fractions. 21/312500. Let's divide numerator and denominator by 21: 1/(312500/21). Let's compute 312500 ÷ 21. 21*14800 = 310,800. 312500 - 310,800 = 1,700. 1,700 ÷21 ≈80.95. So total is approximately 14800 +80.95 ≈14880.95. Therefore, 1/14880.95 ≈0.0000672. So that's approximately 0.0000672. But when I did the long division, I got 0.000672. That's a factor of 10 difference. So where is the mistake?

Wait, when I did the long division, I think I misplaced the decimal. Let me redo the long division step carefully.

Dividing 21 by 312500:

We can write this as 21 ÷ 312500 = (21 ÷ 312500) * (10/10) = (210 ÷ 3125000) * (10/10) = ... but maybe it's better to adjust the divisor to make it a whole number. Alternatively, since 312500 is 5^7 * 2^2 / 2^7? Wait, no. Wait, 312500 is 2^2 * 5^7 divided by 2^2? Wait, no, original denominator is 2^2 *5^7, which is 4*78125=312500. So 21 divided by (2^2 *5^7). To convert this to a decimal, we can adjust the denominator to be a power of 10 by multiplying numerator and denominator by 2^5, as before. So 21*32=672, denominator 10^7. So 672/10,000,000=0.0000672. Therefore, the correct result is 0.0000672. So why did the long division give me 0.000672?

Wait, perhaps I made a mistake in the number of zeros. Let me write it out step by step:

Dividing 21 by 312500:

- 312500 is 3.125 x 10^5. Wait, 312500 is 3125 x 100, which is 5^5 x 2^2 x 5^2 (since 100 is 2^2*5^2). So 5^(5+2)=5^7, and 2^2. So that's correct.

So 21 divided by (2^2 *5^7). To make the denominator 10^7, multiply numerator and denominator by 2^5. So numerator becomes 21*32=672, denominator becomes 2^(2+5)*5^7=2^7*5^7=10^7. Therefore, 672/10^7=0.0000672.

But when I did the long division, I ended up with 0.000672. Let me check the long division again, step by step:

Dividing 21 by 312500:

1. 312500 ) 21.000000

2. 312500 goes into 21.000000 zero times. So we write 0.

3. Add a decimal point and a zero: 210.000000

4. 312500 goes into 210 zero times. Next digit is 0.

5. Add another zero: 2100.000000

6. 312500 goes into 2100 zero times. Next digit is 0.

7. Add another zero: 21000.000000

8. 312500 goes into 21000 zero times. Next digit is 0.

9. Add another zero: 210000.000000

10. Now, 312500 goes into 210000 zero times. Next digit is 0.

11. Add another zero: 2100000.000000

12. Now, 312500 goes into 2100000 six times (6*312500=1,875,000). Subtract 1,875,000 from 2,100,000: remainder 225,000.

13. Bring down the next zero: 2,250,000.

14. 312500 goes into 2,250,000 seven times (7*312500=2,187,500). Subtract: 2,250,000 - 2,187,500 = 62,500.

15. Bring down the next zero: 625,000.

16. 312500 goes into 625,000 two times (2*312500=625,000). Subtract: 625,000 - 625,000 = 0.

17. No more numbers to bring down. So the quotient is 0.000672.

Wait, so according to this, the result is 0.000672. But according to the other method, it's 0.0000672. There's a discrepancy here. Which one is correct?

Wait, let's compute 21 divided by 312500 using a calculator. Let me do that. 21 ÷ 312500. Let's compute:

21 ÷ 312500 = 21 ÷ (3.125 x 10^5) = (21 ÷ 3.125) x 10^-5. 21 ÷ 3.125: 3.125 x 6 = 18.75, 3.125 x 6.72 = 21. So 21 ÷ 3.125 = 6.72. Then, 6.72 x 10^-5 = 0.0000672. So that's 0.0000672. But the long division gave me 0.000672. There's a factor of 10 difference. So where is the mistake?

Ah! Wait, in the long division steps above, I think I messed up the decimal placement. Let me try performing the division again, more carefully.

Dividing 21 by 312500:

- 312500 is 312500.0

- 21.000000 ÷ 312500.0

Since 312500 is larger than 21, we write 0.

- 210.000000 ÷ 312500.0: still 0.

- 2100.000000 ÷ 312500.0: 0.

- 21000.000000 ÷ 312500.0: 0.

- 210000.000000 ÷ 312500.0: 0. So far, 0.0000

- 2100000.000000 ÷ 312500.0: 6 (since 6*312500=1,875,000). Subtract 1,875,000 from 2,100,000: 225,000. Bring down next 0: 2,250,000 ÷ 312500: 7 times (7*312500=2,187,500). Subtract: 2,250,000 - 2,187,500 = 62,500. Bring down next 0: 625,000 ÷ 312500 = 2 times. Subtract 625,000: remainder 0. So the quotient is 0.000672. But according to the other method, it should be 0.0000672. There's a discrepancy here. What's the issue?

Wait, perhaps I made a mistake in the multiplication. Let's check 21*32=672. Correct. 672/10^7=0.0000672. But the long division is giving 0.000672. So there's a factor of 10 difference. Therefore, one of the methods must be wrong. Let me check the long division steps again.

Wait, when I did the long division, I divided 21 by 312500. But 312500 is 3.125 x 10^5. So 21 ÷ (3.125 x 10^5) = (21 ÷ 3.125) x 10^-5. As before, 21 ÷ 3.125 = 6.72. So 6.72 x 10^-5 = 0.0000672. Therefore, the correct result is 0.0000672. So why does the long division give 0.000672? Let me check the long division steps again.

Ah! Wait, in the long division, when I divided 21 by 312500, I considered 21 as 21.000000, but perhaps I missed adding the decimal places correctly. Let me try performing the division step by step with more attention to the decimal placement.

Let's do the division of 21 by 312500 step by step:

1. 312500 ) 21.000000000000

2. 312500 goes into 21.0 zero times. Write 0.

3. 312500 ) 21.000000000000

4. 312500 ) 210.000000000000 (add a decimal point and a zero)

5. 312500 ) 210.000000000000

6. 312500 goes into 210 zero times. Write 0.

7. 312500 ) 2100.000000000000 (add another zero)

8. 312500 ) 2100.000000000000

9. 312500 goes into 2100 zero times. Write 0.

10. 312500 ) 21000.000000000000 (add another zero)

11. 312500 ) 21000.000000000000

12. 312500 goes into 21000 zero times. Write 0.

13. 312500 ) 210000.000000000000 (add another zero)

14. 312500 ) 210000.000000000000

15. 312500 goes into 210000 zero times. Write 0.

16. 312500 ) 2100000.000000000000 (add another zero)

17. 312500 ) 2100000.000000000000

18. 312500 goes into 2100000 six times (6*312500=1,875,000). Subtract 1,875,000 from 2,100,000: 225,000.

19. Bring down the next 0: 2,250,000.

20. 312500 goes into 2,250,000 seven times (7*312500=2,187,500). Subtract: 2,250,000 - 2,187,500 = 62,500.

21. Bring down the next 0: 625,000.

22. 312500 goes into 625,000 two times (2*312500=625,000). Subtract: 625,000 - 625,000 = 0.

23. No more digits to bring down. So the quotient is 0.000672.

Wait, but according to the other method, it should be 0.0000672. There's a contradiction here. Let me check with a calculator. Let's compute 21 divided by 312500:

21 ÷ 312500 = 0.0000672. Yes, that's correct. So the long division must have a mistake. Wait, maybe I messed up the number of zeros. Let me try performing the division again with attention to decimal places.

Alternatively, let's use a different approach. Let's write 21/312500 as (21/3125)/100. 21 divided by 3125: 3125 is 5^5*2^3. Wait, 21/3125 = 21/(5^5*2^3). Multiply numerator and denominator by 2^3 to make denominator 10^3: (21*8)/(5^5*2^3*2^3) = 168/(5^5*10^3). Wait, 5^5 is 3125, so 168/3125 = 0.05376. Then divide by 10^3: 0.05376 / 1000 = 0.00005376. Wait, that's different. Wait, this approach is confusing. Let me compute 21/3125 first.

21 ÷ 3125:

3125 goes into 21 zero times. Add decimal point and zeros: 210 ÷3125=0.0. 210 ÷3125=0.0. 2100 ÷3125=0.00. 21000 ÷3125=0.000. 210000 ÷3125=0.0000. 2100000 ÷3125=0.00000. 21000000 ÷3125=0.000000. Wait, this is getting too confusing. Let me use a calculator for 21 ÷ 3125:

21 ÷ 3125 = 0.00672. Then divide by 100 (since we had 21/312500 = (21/3125)/100), so 0.00672 ÷ 100 = 0.0000672. Therefore, the correct result is 0.0000672. Therefore, the long division method must have an error in the steps. Therefore, the correct decimal is 0.0000672. So the initial long division steps were incorrect due to a miscalculation in the number of decimal places. Therefore, the correct answer is 0.0000672, which matches the other methods.

Therefore, the final answer is 0.0000672.
</execution_4.3>
</Execution_4>

<Final_answer>
Therefore, the terminating decimal equivalent of $\frac{21}{2^2 \cdot 5^7}$ is $\boxed{0.0000672}$.
</Final_answer>
</think>