Question

If \( \frac{1}{(2^{11})(5^{17})} \) is expressed as a terminating decimal, how many nonzero digits will the decimal have?

• A. One
• B. Two
• C. Four
• D. Six
• E. Eleven

Hint:

For a fraction to be a terminating decimal, its denominator, when reduced, must only have the prime factors 2 and 5.

Answer 1

The Correct Option is A

Step 1: Understand the problem.
We are asked to express \( \frac{1}{(2^{11})(5^{17})} \) as a terminating decimal and determine how many nonzero digits the decimal will have.
Step 2: Simplify the expression.
To determine if the decimal terminates, we must first check if the denominator can be factored into powers of 2 and 5, which are the only primes that allow a terminating decimal when used as a denominator.
We have: \[ (2^{11})(5^{17}) = 2^{11} \times 5^{11} \times 5^6 = (2 \times 5)^{11} \times 5^6 = 10^{11} \times 5^6 \] Thus, the denominator is \( 10^{11} \times 5^6 \).
Step 3: Express as a decimal.
The factor of \( 10^{11} \) ensures that the decimal will terminate. The factor \( 5^6 \) will give us a finite number of nonzero digits, so the decimal will have a total of **1 nonzero digit**.
Final Answer: \[ \boxed{\text{The correct answer is (A) One.}} \]