Evaluate. Write your answer as a fraction in simplest form ((4)/(5))^(3)
(A) 4/5 (B) 5/4 (C) 64/125 (D) 125/64
The Correct Answer and Explanation is:
The problem asks to evaluate (\left( \frac{4}{5} \right)^3) and express the result as a fraction in its simplest form.
Step 1: Understanding the problem
The expression (\left( \frac{4}{5} \right)^3) means that we are raising the fraction (\frac{4}{5}) to the power of 3. This involves multiplying the fraction by itself three times:
[
\left( \frac{4}{5} \right)^3 = \frac{4}{5} \times \frac{4}{5} \times \frac{4}{5}
]
Step 2: Multiply the fractions
To multiply fractions, multiply the numerators together and multiply the denominators together:
[
\text{Numerator: } 4 \times 4 \times 4 = 64
]
[
\text{Denominator: } 5 \times 5 \times 5 = 125
]
Thus, the result of multiplying the fractions is:
[
\frac{64}{125}
]
Step 3: Simplifying the result
The fraction (\frac{64}{125}) is already in its simplest form because 64 and 125 have no common factors other than 1. 64 is a power of 2 ((2^6)) and 125 is a power of 5 ((5^3)), so they have no common prime factors.
Step 4: Answer choices
Now, let’s look at the answer choices:
- (A) (\frac{4}{5})
- (B) (\frac{5}{4})
- (C) (\frac{64}{125})
- (D) (\frac{125}{64})
Since we calculated that (\left( \frac{4}{5} \right)^3 = \frac{64}{125}), the correct answer is (C) (\frac{64}{125}).
Conclusion
The correct answer is (C) (\frac{64}{125}).