Answer :
To solve the problem, we need to determine which of the given options is equal to the expression [tex]\((\frac{4}{5})^6\)[/tex].
Let's evaluate each option:
A. [tex]\(\frac{4^6}{5}\)[/tex]
This option represents raising only the numerator to the power of 6, while the denominator remains [tex]\(5\)[/tex]. Since we want both the numerator and denominator raised to the sixth power, this option is incorrect.
B. [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex]
This option means multiplying the fraction [tex]\(\frac{4}{5}\)[/tex] by 6. This does not match the operation of raising the whole fraction to the sixth power, so this option is incorrect.
C. [tex]\(\frac{24}{30}\)[/tex]
This fraction simplifies to [tex]\(\frac{4}{5}\)[/tex], but it's not [tex]\((\frac{4}{5})^6\)[/tex]. Therefore, this option is incorrect.
D. [tex]\(\frac{4^6}{5^6}\)[/tex]
This option involves raising both the numerator, [tex]\(4\)[/tex], and the denominator, [tex]\(5\)[/tex], to the sixth power, which is exactly what [tex]\((\frac{4}{5})^6\)[/tex] means. As a result, this option is correct.
Thus, the correct answer is option D: [tex]\(\frac{4^6}{5^6}\)[/tex].
Let's evaluate each option:
A. [tex]\(\frac{4^6}{5}\)[/tex]
This option represents raising only the numerator to the power of 6, while the denominator remains [tex]\(5\)[/tex]. Since we want both the numerator and denominator raised to the sixth power, this option is incorrect.
B. [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex]
This option means multiplying the fraction [tex]\(\frac{4}{5}\)[/tex] by 6. This does not match the operation of raising the whole fraction to the sixth power, so this option is incorrect.
C. [tex]\(\frac{24}{30}\)[/tex]
This fraction simplifies to [tex]\(\frac{4}{5}\)[/tex], but it's not [tex]\((\frac{4}{5})^6\)[/tex]. Therefore, this option is incorrect.
D. [tex]\(\frac{4^6}{5^6}\)[/tex]
This option involves raising both the numerator, [tex]\(4\)[/tex], and the denominator, [tex]\(5\)[/tex], to the sixth power, which is exactly what [tex]\((\frac{4}{5})^6\)[/tex] means. As a result, this option is correct.
Thus, the correct answer is option D: [tex]\(\frac{4^6}{5^6}\)[/tex].
