College

Which of the following is equal to the fraction below?

[tex]\[

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

\][/tex]

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

B. [tex]\(\frac{24}{30}\)[/tex]

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

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

Answer :

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

1. Original Expression:
The original expression is [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].

2. Option A: [tex]\(\frac{4^6}{5^6}\)[/tex]
- When you raise a fraction to a power, both the numerator and the denominator are raised to that power.
- [tex]\(\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}\)[/tex].
- This matches the original expression exactly.

3. Option B: [tex]\(\frac{24}{30}\)[/tex]
- Simplifying [tex]\(\frac{24}{30}\)[/tex] gives [tex]\(\frac{4}{5}\)[/tex], which is not the same as raising to the 6th power.
- Therefore, this does not equal [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].

4. Option C: [tex]\(\frac{4^6}{5}\)[/tex]
- This representation does not raise the denominator to the 6th power, so it does not match the original expression.

5. Option D: [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex]
- Multiplying [tex]\(6\)[/tex] by [tex]\(\frac{4}{5}\)[/tex] does not involve an exponent of 6.
- This is different from the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].

After comparing each option, the only expression that accurately represents [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is Option A: [tex]\(\frac{4^6}{5^6}\)[/tex].