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Answer :
To solve the problem of finding which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], we need to evaluate each of the given choices:
1. Option A: [tex]\(\frac{4^6}{5^6}\)[/tex]
When you raise a fraction to a power, you raise both the numerator and the denominator to that power. So:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
This matches exactly with Option A.
2. Option B: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This option suggests multiplying the fraction by 6, not raising it to a power. The calculation is:
[tex]\[
6 \cdot \frac{4}{5} = \frac{24}{5}
\][/tex]
This is clearly not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
3. Option C: [tex]\(\frac{24}{30}\)[/tex]
Simplifying [tex]\(\frac{24}{30}\)[/tex] gives:
[tex]\[
\frac{24}{30} = \frac{4 \cdot 6}{5 \cdot 6} = \frac{4}{5}
\][/tex]
This is just the fraction itself and not raised to any power. So, it's not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
4. Option D: [tex]\(\frac{4^6}{5}\)[/tex]
Here, only the numerator is raised to the power 6, not the denominator:
[tex]\[
\frac{4^6}{5}
\][/tex]
This does not represent [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], as both the numerator and denominator should be raised to the power 6.
Given these evaluations, Option A: [tex]\(\frac{4^6}{5^6}\)[/tex] is the correct choice, as it is equivalent to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
1. Option A: [tex]\(\frac{4^6}{5^6}\)[/tex]
When you raise a fraction to a power, you raise both the numerator and the denominator to that power. So:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
This matches exactly with Option A.
2. Option B: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This option suggests multiplying the fraction by 6, not raising it to a power. The calculation is:
[tex]\[
6 \cdot \frac{4}{5} = \frac{24}{5}
\][/tex]
This is clearly not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
3. Option C: [tex]\(\frac{24}{30}\)[/tex]
Simplifying [tex]\(\frac{24}{30}\)[/tex] gives:
[tex]\[
\frac{24}{30} = \frac{4 \cdot 6}{5 \cdot 6} = \frac{4}{5}
\][/tex]
This is just the fraction itself and not raised to any power. So, it's not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
4. Option D: [tex]\(\frac{4^6}{5}\)[/tex]
Here, only the numerator is raised to the power 6, not the denominator:
[tex]\[
\frac{4^6}{5}
\][/tex]
This does not represent [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], as both the numerator and denominator should be raised to the power 6.
Given these evaluations, Option A: [tex]\(\frac{4^6}{5^6}\)[/tex] is the correct choice, as it is equivalent to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
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