High School

Which of the following is equal to the fraction below?

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

A. \[ \frac{24}{30} \]

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

C. \[ \frac{4^8}{5} \]

D. \[ 6 \cdot \left(\frac{4}{5}\right) \]

Answer :

To determine which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], let's analyze each choice:

1. Expression: [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is calculated by raising both the numerator and the denominator to the power of 6. So, it can be represented as:
[tex]\[
\frac{4^6}{5^6}
\][/tex]

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

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 6:
[tex]\[
\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}
\][/tex]
[tex]\(\frac{4}{5}\)[/tex] does not equal [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].

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

As previously mentioned, the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is equivalent to this expression:
[tex]\[
\frac{4^6}{5^6}
\][/tex]
They are equal.

4. Option C: [tex]\(\frac{4^8}{5}\)[/tex]

This option is not equivalent to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] since in [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], both the numerator and the denominator are raised to the same power.

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

This represents multiplying [tex]\(\frac{4}{5}\)[/tex] by 6:
[tex]\[
6 \cdot \frac{4}{5} = \frac{24}{5}
\][/tex]
This is clearly not the same as [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].

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