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Answer :
To determine which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], we need to understand the expression and see how we can simplify or rewrite it.
The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means raising the fraction [tex]\(\frac{4}{5}\)[/tex] to the 6th power. This involves raising both the numerator (4) and the denominator (5) to the 6th power separately.
1. Calculate [tex]\(4^6\)[/tex]:
- [tex]\(4^6 = 4096\)[/tex]
2. Calculate [tex]\(5^6\)[/tex]:
- [tex]\(5^6 = 15625\)[/tex]
This means:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6} = \frac{4096}{15625}
\][/tex]
Now, let's compare this with the given answer choices:
A. [tex]\(\frac{4^6}{5}\)[/tex] - This is not correct as it only raises the numerator to the 6th power, not the entire fraction.
B. [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex] - This represents multiplying the fraction by 6, not raising the fraction to the 6th power, so it is not correct.
C. [tex]\(\frac{24}{30}\)[/tex] - This is a simplified fraction that is equivalent to [tex]\(\frac{4}{5}\)[/tex], certainly not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
D. [tex]\(\frac{4^6}{5^6}\)[/tex] - This matches our calculated result for [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
The correct answer is D. [tex]\(\frac{4^6}{5^6}\)[/tex].
The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means raising the fraction [tex]\(\frac{4}{5}\)[/tex] to the 6th power. This involves raising both the numerator (4) and the denominator (5) to the 6th power separately.
1. Calculate [tex]\(4^6\)[/tex]:
- [tex]\(4^6 = 4096\)[/tex]
2. Calculate [tex]\(5^6\)[/tex]:
- [tex]\(5^6 = 15625\)[/tex]
This means:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6} = \frac{4096}{15625}
\][/tex]
Now, let's compare this with the given answer choices:
A. [tex]\(\frac{4^6}{5}\)[/tex] - This is not correct as it only raises the numerator to the 6th power, not the entire fraction.
B. [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex] - This represents multiplying the fraction by 6, not raising the fraction to the 6th power, so it is not correct.
C. [tex]\(\frac{24}{30}\)[/tex] - This is a simplified fraction that is equivalent to [tex]\(\frac{4}{5}\)[/tex], certainly not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
D. [tex]\(\frac{4^6}{5^6}\)[/tex] - This matches our calculated result for [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
The correct answer is D. [tex]\(\frac{4^6}{5^6}\)[/tex].
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