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 and only if the denominator, after simplifying the fraction, has no prime factors other than 2 and 5. In this case, the denominator is already 2² times 5⁷, which are both 2 and 5. So that checks out—it should be a terminating decimal. 

But how do I actually convert it? Let me recall the steps. I think the key is to make the denominator a power of 10, since 10 is 2 times 5. So if I can adjust the fraction so that the denominator becomes 10 raised to some power, then the numerator will give me the decimal. 

Right, so the denominator here is 2² * 5⁷. To make this a power of 10, I need to have equal numbers of 2s and 5s because 10 = 2*5. So the denominator has more 5s than 2s. There are 2 twos and 7 fives. So I need to multiply both the numerator and the denominator by enough 2s to make the number of 2s equal to the number of 5s. 

Wait, actually, since the denominator has 2² and 5⁷, the number of 2s is 2 and 5s is 7. To make them equal, I need to add 5 more 2s to the denominator. That way, both will have 7 2s and 7 5s, making the denominator 10⁷. 

But since I can't just add 2s to the denominator without multiplying the numerator as well, I need to multiply both numerator and denominator by 2⁵. Let me check that. 

Original denominator: 2² * 5⁷. Multiply by 2⁵: 2² * 2⁵ = 2⁷, and 5⁷ remains. So the new denominator is 2⁷ * 5⁷ = (2*5)⁷ = 10⁷. 

So if I multiply the numerator 21 by 2⁵, which is 32, then the new fraction becomes (21 * 32) / 10⁷. Let me compute 21 * 32. 

21 times 30 is 630, and 21 times 2 is 42, so 630 + 42 = 672. So the fraction is 672 / 10⁷. 

Now, dividing by 10⁷ moves the decimal point seven places to the left. So 672 divided by 10⁷ is 0.0000672. Wait, but let me count the decimal places. 

Starting from 672, moving the decimal seven places left: 672.0 becomes 0.6720 after one place, 0.06720 after two, 0.006720 after three, 0.0006720 after four, 0.00006720 after five, 0.000006720 after six, and 0.000000672 after seven. Wait, that's only six decimal places. Hmm, maybe I made a mistake here. 

Wait, 672 divided by 10⁷. Let's think of 10⁷ as 10,000,000. So 672 divided by 10,000,000. 

So 672 ÷ 10,000,000. Let me write that as a decimal. 

672 divided by 10,000,000. Since 10,000,000 is 10^7, moving the decimal seven places to the left. Starting with 672.0, moving the decimal one place left gives 67.2, then 6.72, then 0.672, 0.0672, 0.00672, 0.000672, 0.0000672. Wait, but that's seven places. Wait, 672.0 divided by 10^1 is 67.2, divided by 10^2 is 6.72, divided by 10^3 is 0.672, 10^4 is 0.0672, 10^5 is 0.00672, 10^6 is 0.000672, and 10^7 is 0.0000672. So yes, that's correct. 

So 672 divided by 10^7 is 0.0000672. But wait, let me check that calculation again. 

Alternatively, 21/(2²*5⁷) = 21/(4*78125) = 21/312500. Let me compute 21 divided by 312500. 

Alternatively, 21 divided by 312500. Let me see. 312500 x 0.0000672 = 312500 * 6.72e-5. Let's compute 312500 * 6.72e-5. 

First, 312500 * 6.72e-5. 312500 * 6.72e-5 = 312500 * 6.72 / 100000. 

Compute 312500 * 6.72 first. 

312500 * 6 = 1,875,000. 

312500 * 0.72 = 312500 * 0.7 + 312500 * 0.02 = 218,750 + 6,250 = 225,000. 

So total is 1,875,000 + 225,000 = 2,100,000. 

Then divide by 100,000: 2,100,000 / 100,000 = 21. 

Yes, that checks out. So 312500 * 0.0000672 = 21. So the decimal is correct. 

Therefore, 21/(2²*5⁷) = 0.0000672. 

But let me verify once more. Let's compute the original fraction step by step. 

21 divided by (2²*5⁷) = 21/(4*78125) = 21/312500. 

Now, 21 divided by 312500. Let's do this division. 

312500 goes into 21 zero times. Add a decimal point and zeros. 

312500 into 2100000 (adding four zeros after 21 to make it 210000). So 312500 * 6 = 1,875,000. Subtract 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. Subtract: 625,000 - 625,000 = 0. So the division is exact here. 

So the decimal is 0.0000672. 

Wait, let me count the decimal places: 0.0000672. That's 7 decimal places. Let's confirm. 

Starting from the decimal point: 

0.0 (1st place) 

0.00 (2nd) 

0.000 (3rd) 

0.0000 (4th) 

0.00000 (5th) 

0.000000 (6th) 

0.0000000 (7th). Wait, but the number is 0.0000672. So after the decimal, it's 0,0,0,0,6,7,2. That's seven decimal places. 

Yes, so the decimal is 0.0000672. 

But let me check once more with another method. Let's convert the original fraction to decimal by simplifying the denominator. 

Original fraction: 21/(2²*5⁷). 

We can write this as (21 * 2⁵) / (2²*5⁷*2⁵) = (21*32)/(2⁷*5⁷) = 672/10⁷ = 0.0000672. 

Yes, that's the same result. 

Alternatively, 21/(4*78125) = 21/312500. If I divide 21 by 312500, moving the decimal point seven places to the left. 

But 21 divided by 312500 is the same as 21 divided by 3.125 x 10^5, which is (21 / 3.125) x 10^-5. 

21 divided by 3.125. Let's compute that. 

3.125 goes into 21 how many times? 

3.125 * 6 = 18.75 

3.125 * 6.72 = 21. So 6.72 x 10^-5. Wait, that's 0.0000672. 

Yes, same result. 

Therefore, the decimal is 0.0000672. 

Wait, but let me check if there's a simpler way. Let's see. 

Another approach: since the denominator is 2²*5⁷, we can adjust the numerator and denominator to make the denominator a power of 10. 

We need to multiply numerator and denominator by 5^(7-2) = 5^5 to make the denominator 2^2*5^7*5^5 = 2^2*5^12. Wait, no, that's not helpful. 

Wait, maybe I need to adjust so that the denominator has equal exponents of 2 and 5. 

The denominator has 2²*5⁷. To make them equal, we need to have 5s and 2s equal. So since there are more 5s, we need to multiply by 2^(7-2) = 2^5 to make the denominator 2^(2+5)*5^7 = 2^7*5^7 = 10^7. 

So multiplying numerator and denominator by 2^5: 

Numerator becomes 21*2^5 = 21*32 = 672 

Denominator becomes 10^7 = 10,000,000 

So 672 / 10,000,000 = 0.0000672. 

Same result. 

Therefore, I think the answer is 0.0000672. 

Wait, but let me check once again with another method. Let's use prime factorization. 

The original fraction is 21/(2²*5⁷). 21 is 3*7. So the numerator is 3*7, denominator is 2²*5⁷. 

To convert to a decimal, we need the denominator to be a power of 10. The number of decimal places will be the maximum of the exponents of 2 and 5 in the denominator. Wait, no, the number of decimal places is the maximum of the exponents. Wait, actually, when converting to a decimal, the number of decimal places needed is determined by the denominator when written as 10^n. 

But since the denominator is 2²*5⁷, to make it 10^n, we need to have both exponents equal to the maximum of 2 and 7, which is 7. So we multiply numerator and denominator by 2^(7-2) = 2^5 to get denominator 10^7. 

So numerator becomes 21*2^5 = 672, denominator 10^7. So 672/10^7 is 0.0000672. 

Yes, that's consistent. 

Therefore, after multiple methods and checks, I believe the terminating decimal is 0.0000672. 

But let me confirm once more by actually performing the long division of 21 by 312500. 

Let me write this out step by step. 

Dividing 21 by 312500. 

312500 goes into 21 zero times. Add a decimal point and zeros: 21.000000...

312500 into 2100000 (21.000000 becomes 2100000 after moving decimal 5 places). 

312500 * 6 = 1,875,000. Subtract: 2,100,000 - 1,875,000 = 225,000. 

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

312500 * 7 = 2,187,500. Subtract: 2,250,000 - 2,187,500 = 62,500. 

Bring down the next 0: 625,000. 

312500 * 2 = 625,000. Subtract: 625,000 - 625,000 = 0. 

Bring down the next 0: 0.0. 

So the division gives 0.0000672. 

Yes, that's exactly the decimal. 

Therefore, after all these steps and checks, I'm confident that the correct answer is 0.0000672. 

**Final Answer**
The fraction $\frac{21}{2^2 \cdot 5^7}$ converted to a terminating decimal is \boxed{0.0000672}.
Therefore, $\frac{21}{2^2 \cdot 5^7}$ as a terminating decimal is $\boxed{0.0000672}$.
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