Answer :
To solve the problem, we need to determine which of the given options is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Let's examine each option step by step:
1. Option A: [tex]\(\frac{4^6}{5}\)[/tex]
This expression means raising 4 to the 6th power and then dividing by 5. However, [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] involves both the numerator and the denominator being raised to the 6th power, so this option does not match.
2. Option B: [tex]\(\frac{24}{30}\)[/tex]
This fraction simplifies to [tex]\(\frac{4}{5}\)[/tex] when reduced, but we need [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], which is [tex]\(\frac{4^6}{5^6}\)[/tex], not just [tex]\(\frac{4}{5}\)[/tex]. Thus, this option is incorrect.
3. Option C: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This option multiplies [tex]\(\frac{4}{5}\)[/tex] by 6, but we need [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], which involves both the numerator and the denominator being raised to the 6th power. Therefore, this option is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
4. Option D: [tex]\(\frac{4^6}{5^6}\)[/tex]
This option represents [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] because when you raise [tex]\(\frac{4}{5}\)[/tex] to the power of 6, you raise both 4 and 5 to the power of 6 separately. This matches exactly what [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is, so this option is correct.
Thus, the correct answer is Option D: [tex]\(\frac{4^6}{5^6}\)[/tex].
Let's examine each option step by step:
1. Option A: [tex]\(\frac{4^6}{5}\)[/tex]
This expression means raising 4 to the 6th power and then dividing by 5. However, [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] involves both the numerator and the denominator being raised to the 6th power, so this option does not match.
2. Option B: [tex]\(\frac{24}{30}\)[/tex]
This fraction simplifies to [tex]\(\frac{4}{5}\)[/tex] when reduced, but we need [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], which is [tex]\(\frac{4^6}{5^6}\)[/tex], not just [tex]\(\frac{4}{5}\)[/tex]. Thus, this option is incorrect.
3. Option C: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This option multiplies [tex]\(\frac{4}{5}\)[/tex] by 6, but we need [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], which involves both the numerator and the denominator being raised to the 6th power. Therefore, this option is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
4. Option D: [tex]\(\frac{4^6}{5^6}\)[/tex]
This option represents [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] because when you raise [tex]\(\frac{4}{5}\)[/tex] to the power of 6, you raise both 4 and 5 to the power of 6 separately. This matches exactly what [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is, so this option is correct.
Thus, the correct answer is Option D: [tex]\(\frac{4^6}{5^6}\)[/tex].
