College

Which of the following is equal to the fraction below?

[tex]\left(\frac{4}{5}\right)^6[/tex]

A. [tex]\frac{24}{30}[/tex]
B. [tex]6 \cdot\left(\frac{4}{5}\right)[/tex]
C. [tex]\frac{4^6}{5^6}[/tex]
D. [tex]\frac{4^6}{5}[/tex]

Answer :

To solve this question, we need to understand how to work with powers of fractions:

1. Understand the Expression: We are looking at [tex]\((\frac{4}{5})^6\)[/tex]. When we raise a fraction to a power, both the numerator and the denominator are raised to that power.

2. Apply the Rule: For [tex]\((\frac{4}{5})^6\)[/tex], this means the numerator 4 and the denominator 5 are each raised to the power of 6. This gives us:

[tex]\[
(\frac{4}{5})^6 = \frac{4^6}{5^6}
\][/tex]

3. Match with the Given Options: From the given options, we are looking for one that matches [tex]\(\frac{4^6}{5^6}\)[/tex].

- Option A: [tex]\(\frac{24}{30}\)[/tex] - This is a simplified fraction and does not match [tex]\(\frac{4^6}{5^6}\)[/tex].
- Option B: [tex]\(6 \cdot(\frac{4}{5})\)[/tex] - This involves multiplication by 6, not raising to the sixth power.
- Option C: [tex]\(\frac{4^6}{5^6}\)[/tex] - This is exactly what we calculated.
- Option D: [tex]\(\frac{4^6}{5}\)[/tex] - This does not raise 5 to the sixth power.

Therefore, the expression [tex]\((\frac{4}{5})^6\)[/tex] is equal to [tex]\(\frac{4^6}{5^6}\)[/tex], which corresponds to option C.