College

Which of the following is equal to the expression 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 the problem of identifying which option is equal to the expression [tex]\((\frac{4}{5})^6\)[/tex], let's look at each of the given choices:

1. Option A: [tex]\(\frac{24}{30}\)[/tex]

- Simplify [tex]\(\frac{24}{30}\)[/tex]:
[tex]\[
\frac{24}{30} = \frac{4 \times 6}{5 \times 6} = \frac{4}{5}
\][/tex]
- [tex]\(\frac{4}{5}\)[/tex] is not equal to [tex]\((\frac{4}{5})^6\)[/tex].

2. Option B: [tex]\(6 \cdot \frac{4}{5}\)[/tex]

- Compute [tex]\(6 \cdot \frac{4}{5}\)[/tex]:
[tex]\[
6 \cdot \frac{4}{5} = \frac{24}{5}
\][/tex]
- [tex]\(\frac{24}{5}\)[/tex] is not equal to [tex]\((\frac{4}{5})^6\)[/tex].

3. Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]

- Notice that [tex]\(\frac{4^6}{5^6}\)[/tex] is exactly [tex]\((\frac{4}{5})^6\)[/tex] because:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
- This option matches the given expression [tex]\((\frac{4}{5})^6\)[/tex].

4. Option D: [tex]\(\frac{4^6}{5}\)[/tex]

- Calculate if [tex]\(\frac{4^6}{5}\)[/tex] is equal to [tex]\((\frac{4}{5})^6\)[/tex]:
- Since it isn't structured with the same base and power, this option does not match [tex]\((\frac{4}{5})^6\)[/tex].

Based on the findings, the expression [tex]\((\frac{4}{5})^6\)[/tex] is equal to [tex]\(\frac{4^6}{5^6}\)[/tex]. Therefore, Option C: [tex]\(\frac{4^6}{5^6}\)[/tex] is correct.