High School

Which of the following is equal to the fraction below?

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

A. [tex]6 \cdot\left(\frac{4}{5}\right)[/tex]

B. [tex]\frac{4^6}{5}[/tex]

C. [tex]\frac{4^6}{5^6}[/tex]

D. [tex]\frac{24}{30}[/tex]

Answer :

Let's solve the problem to find which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].

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

The fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] evaluates approximately to 0.262144.

2. Check each option:

Option A: [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex]

[tex]\(6 \times \frac{4}{5} = \frac{24}{5} = 4.8\)[/tex]

This is not approximately 0.262144, so Option A is incorrect.

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

First compute [tex]\(4^6 = 4096\)[/tex], then divide by 5:

[tex]\(\frac{4096}{5} = 819.2\)[/tex]

This is not approximately 0.262144, so Option B is incorrect.

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

Again, [tex]\(4^6 = 4096\)[/tex] and [tex]\(5^6 = 15625\)[/tex]:

[tex]\(\frac{4096}{15625} \approx 0.262144\)[/tex]

This is approximately equal to 0.262144, making Option C correct.

Option D: [tex]\(\frac{24}{30}\)[/tex]

Simplifying this fraction: [tex]\(\frac{24}{30} = \frac{4}{5} = 0.8\)[/tex]

This is not approximately 0.262144, so Option D is incorrect.

3. Conclusion:

The option that is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is Option C: [tex]\(\frac{4^6}{5^6}\)[/tex].