Walkabout wrote:
If \(d=\frac{1}{(2^3)(5^7)}\) is expressed as a terminating decimal, how many nonzero digits will d have?
(A) One
(B) Two
(C) Three
(D) Seven
(E) Ten
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
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