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Answer :
Alright, let's take a closer look at this problem and work through it step-by-step.
We need to find an equivalent expression for the fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
1. Understanding Exponentiation with Fractions:
When we raise a fraction to a power, both the numerator and the denominator are raised to that power. For the fraction [tex]\(\left(\frac{4}{5}\right)\)[/tex], raising it to the 6th power means we do this:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
2. Matching with the Choices:
Let's compare this result with the given options:
- Option A: [tex]\(\frac{4^6}{5}\)[/tex]: This only has the numerator, [tex]\(4^6\)[/tex], raised to the 6th power.
- Option B: [tex]\(\frac{24}{30}\)[/tex]: This fraction simplifies to [tex]\(\frac{4}{5}\)[/tex], but is not raised to any power.
- Option C: [tex]\(6 \bullet\left(\frac{4}{5}\right)\)[/tex]: This is multiplying [tex]\(\frac{4}{5}\)[/tex] by 6, not raising it to the 6th power.
- Option D: [tex]\(\frac{4^6}{5^6}\)[/tex]: This matches our expression exactly. Both the numerator and the denominator are raised to the 6th power.
Thus, the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is equal to [tex]\(\frac{4^6}{5^6}\)[/tex].
Therefore, the correct answer is Option D: [tex]\(\frac{4^6}{5^6}\)[/tex].
We need to find an equivalent expression for the fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
1. Understanding Exponentiation with Fractions:
When we raise a fraction to a power, both the numerator and the denominator are raised to that power. For the fraction [tex]\(\left(\frac{4}{5}\right)\)[/tex], raising it to the 6th power means we do this:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
2. Matching with the Choices:
Let's compare this result with the given options:
- Option A: [tex]\(\frac{4^6}{5}\)[/tex]: This only has the numerator, [tex]\(4^6\)[/tex], raised to the 6th power.
- Option B: [tex]\(\frac{24}{30}\)[/tex]: This fraction simplifies to [tex]\(\frac{4}{5}\)[/tex], but is not raised to any power.
- Option C: [tex]\(6 \bullet\left(\frac{4}{5}\right)\)[/tex]: This is multiplying [tex]\(\frac{4}{5}\)[/tex] by 6, not raising it to the 6th power.
- Option D: [tex]\(\frac{4^6}{5^6}\)[/tex]: This matches our expression exactly. Both the numerator and the denominator are raised to the 6th power.
Thus, the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is equal to [tex]\(\frac{4^6}{5^6}\)[/tex].
Therefore, the correct answer is Option D: [tex]\(\frac{4^6}{5^6}\)[/tex].
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