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Answer :
To solve the problem of identifying which option is equal to the expression [tex]\((\frac{4}{5})^6\)[/tex], let's look at each of the given choices:
1. Option A: [tex]\(\frac{24}{30}\)[/tex]
- Simplify [tex]\(\frac{24}{30}\)[/tex]:
[tex]\[
\frac{24}{30} = \frac{4 \times 6}{5 \times 6} = \frac{4}{5}
\][/tex]
- [tex]\(\frac{4}{5}\)[/tex] is not equal to [tex]\((\frac{4}{5})^6\)[/tex].
2. Option B: [tex]\(6 \cdot \frac{4}{5}\)[/tex]
- Compute [tex]\(6 \cdot \frac{4}{5}\)[/tex]:
[tex]\[
6 \cdot \frac{4}{5} = \frac{24}{5}
\][/tex]
- [tex]\(\frac{24}{5}\)[/tex] is not equal to [tex]\((\frac{4}{5})^6\)[/tex].
3. Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
- Notice that [tex]\(\frac{4^6}{5^6}\)[/tex] is exactly [tex]\((\frac{4}{5})^6\)[/tex] because:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
- This option matches the given expression [tex]\((\frac{4}{5})^6\)[/tex].
4. Option D: [tex]\(\frac{4^6}{5}\)[/tex]
- Calculate if [tex]\(\frac{4^6}{5}\)[/tex] is equal to [tex]\((\frac{4}{5})^6\)[/tex]:
- Since it isn't structured with the same base and power, this option does not match [tex]\((\frac{4}{5})^6\)[/tex].
Based on the findings, the expression [tex]\((\frac{4}{5})^6\)[/tex] is equal to [tex]\(\frac{4^6}{5^6}\)[/tex]. Therefore, Option C: [tex]\(\frac{4^6}{5^6}\)[/tex] is correct.
1. Option A: [tex]\(\frac{24}{30}\)[/tex]
- Simplify [tex]\(\frac{24}{30}\)[/tex]:
[tex]\[
\frac{24}{30} = \frac{4 \times 6}{5 \times 6} = \frac{4}{5}
\][/tex]
- [tex]\(\frac{4}{5}\)[/tex] is not equal to [tex]\((\frac{4}{5})^6\)[/tex].
2. Option B: [tex]\(6 \cdot \frac{4}{5}\)[/tex]
- Compute [tex]\(6 \cdot \frac{4}{5}\)[/tex]:
[tex]\[
6 \cdot \frac{4}{5} = \frac{24}{5}
\][/tex]
- [tex]\(\frac{24}{5}\)[/tex] is not equal to [tex]\((\frac{4}{5})^6\)[/tex].
3. Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
- Notice that [tex]\(\frac{4^6}{5^6}\)[/tex] is exactly [tex]\((\frac{4}{5})^6\)[/tex] because:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
- This option matches the given expression [tex]\((\frac{4}{5})^6\)[/tex].
4. Option D: [tex]\(\frac{4^6}{5}\)[/tex]
- Calculate if [tex]\(\frac{4^6}{5}\)[/tex] is equal to [tex]\((\frac{4}{5})^6\)[/tex]:
- Since it isn't structured with the same base and power, this option does not match [tex]\((\frac{4}{5})^6\)[/tex].
Based on the findings, the expression [tex]\((\frac{4}{5})^6\)[/tex] is equal to [tex]\(\frac{4^6}{5^6}\)[/tex]. Therefore, Option C: [tex]\(\frac{4^6}{5^6}\)[/tex] is correct.
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