If d=1/(2^3*5^7) is expressed as a terminating decimal, how many nonze

Since actually dividing 1/(2^3*5^7) would be time consuming, we want to manipulate d so that we are working with a cleaner denominator. The easiest way to do that is to multiply d by a value that will produce a perfect power of 10 in the denominator. This means that the number of 2s in the denominator will equal the number of 5s in the denominator.

Thus, we can multiply 1/(2^3*5^7) by 2^4/2^4. This gives us:

2^4/(2^7*5^7)

2^4/10^7

16/10^7

16/10,000,000

We can stop here because we know that the 10,000,000 in the denominator means to move the decimal place after the 16 seven places to the left. The final value of d will be 0.0000016. Note that the division of 16 by 10,000,000 did not produce any additional non-zero digits. Thus d has 2 non-zero digits.

Answer is B.

1/2^3*5^7 = 2^-3*5^-7 =.002 * .0000007. So there are 2 non zero digits!!

Unfortunately this is not correct:

\(2^{-3}=\frac{1}{8}=0.125\) not 0.002, which is 2/10^3 and \(5^{-7}=\frac{1}{78,125}=0.0000128\) not 0.0000007, which is 7/10^7.

Hope it helps.

I have seen couple of more problem like this. One thing is still not clear to me. When you multiply whole denominator by 2^4 why is 5^7 getting ignored? Shouldn't 2^4 multiply both 2^3 as well as 5^7?

Thanks

Frankly, the red part does not make any sense...

The denominator is \(2^7*5^7\). Multiply it by \(2^4\). What do you get?

What is it that you saw that indicated you should multiply by 2^4. Just looking at the problem that never occurred to me and I'd like to understand why it did to you.

We need to multiply by 2^4/2^4 in order to convert the denominator to the base of 10 and then to convert the fraction into the decimal form: 0.xxxx.

Similar questions to practice:
https://gmatclub.com/forum/if-t-1-2-9-5- ... 29447.html
https://gmatclub.com/forum/if-d-1-2-3-5- ... 28457.html

Hope this helps.

\(d\) = \(\frac{1}{(2^3*5^7)}\)

=>\(d\) = \(\frac{1}{(2^3*5^3*5^4)}\)

=>\(d\) = \(\frac{1}{(10^3*5^4)}\)

\(\frac{1}{5}\) = \(0.20\)

\(\frac{1}{25}\) = \(\frac{0.20}{5}\) => \(0.04\)

\(\frac{1}{125}\) = \(\frac{0.04}{5}\) => \(0.008\)

\(\frac{1}{625}\) = \(\frac{0.008}{5}\) => \(0.0016\)

Hence there will be 2 non zero digits...

Feel free to revert in case of any doubt ( I have used some shortcuts , would love to explain if needed )

We are given:



In the denominator, we have two numbers with different bases and different exponents. We can rewrite those numbers to have same exponents.





\(2^4 = 16\) and \(10^7\) gives us \(7\) decimal places. We can write this as:



We have 2 non-zero digits. The final answer is .

Let's take the fraction \(\frac{1}{(2^3)(5^7)}\) and find an equivalent fraction that has a power of 10 in its denominator.

Why do this?
Well, it's very easy to take a fraction with a power of 10 in its denominator and convert it to a decimal

For example:
\(\frac{33}{1,000}=0.033\)

\(\frac{1891}{1,000,000}=0.001891\)

\(\frac{7}{100,000}=0.00007\)


Okay, let's begin...

Take: \(\frac{1}{(2^3)(5^7)}\)

Multiply numerator and denominator by \(2^4\) to get: \(\frac{2^4}{(2^4)(2^3)(5^7)}\)

Simplify denominator to get: \(\frac{2^4}{(2^7)(5^7)}\)

Rewrite denominator as follows: \(\frac{2^4}{10^7}\) ASIDE: I applied the rule that says \((x^k)(y^k)=(xy)^k\)

Simplify: \(\frac{16}{10,000,000}=0.0000016\)

Answer: B

Cheers,
Brent

Asked: If \(d=\frac{1}{2^3*5^7}\) is expressed as a terminating decimal, how many nonzero digits will d have?

\(d=\frac{1}{2^3*5^7} = \frac{2^4}{2^75^7} = 2^4 * 10^{-7} = 16*10^{-7} = 0.0000016\)

2 non-zero digit (16)

IMO B

Why do we choose 2^4 ?

In the denominator we have 2^3*5^7. If we multiply 2^3*5^7 by 2^4 we get 2^7*5^7 = 10^7. So, we need to multiply the entire fraction by 2^4/2^4 in order to convert the denominator to the base of 10 and then to convert the fraction into the decimal form: 0.xxxx.