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Answer :
To solve the problem [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] and find which option is equal to this value, we can evaluate the expression directly and compare it to the provided options.
1. Calculate [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
This fraction is calculated by raising both the numerator and the denominator to the power of 6.
2. Evaluate each option:
- Option A: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This would equal [tex]\(6 \cdot \frac{4}{5} = \frac{24}{5}\)[/tex]. This does not match our fraction [tex]\(\frac{4^6}{5^6}\)[/tex].
- Option B: [tex]\(\frac{4^6}{5}\)[/tex]
This is not the same as [tex]\(\frac{4^6}{5^6}\)[/tex] because the power of 5 in the denominator is not equal to 6.
- Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
This is the correct evaluation of [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] because both the numerator and the denominator have been raised to the power of 6.
- Option D: [tex]\(\frac{24}{30}\)[/tex]
This simplifies to [tex]\(\frac{4}{5}\)[/tex], which is not equal to [tex]\(\frac{4^6}{5^6}\)[/tex].
3. Conclusion:
The option that represents [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] correctly is Option C: [tex]\(\frac{4^6}{5^6}\)[/tex].
1. Calculate [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
This fraction is calculated by raising both the numerator and the denominator to the power of 6.
2. Evaluate each option:
- Option A: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This would equal [tex]\(6 \cdot \frac{4}{5} = \frac{24}{5}\)[/tex]. This does not match our fraction [tex]\(\frac{4^6}{5^6}\)[/tex].
- Option B: [tex]\(\frac{4^6}{5}\)[/tex]
This is not the same as [tex]\(\frac{4^6}{5^6}\)[/tex] because the power of 5 in the denominator is not equal to 6.
- Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
This is the correct evaluation of [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] because both the numerator and the denominator have been raised to the power of 6.
- Option D: [tex]\(\frac{24}{30}\)[/tex]
This simplifies to [tex]\(\frac{4}{5}\)[/tex], which is not equal to [tex]\(\frac{4^6}{5^6}\)[/tex].
3. Conclusion:
The option that represents [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] correctly is Option C: [tex]\(\frac{4^6}{5^6}\)[/tex].
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