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Answer :
To solve this problem, we need to determine which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Let's evaluate the given options:
1. Option A: [tex]\(\frac{24}{30}\)[/tex]
Simplify [tex]\(\frac{24}{30}\)[/tex]:
[tex]\[
\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}
\][/tex]
This is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
2. Option B: [tex]\(\frac{4^6}{5}\)[/tex]
This expression is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] because the denominator is not raised to the 6th power.
3. Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
This expression represents [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] because both the numerator and the denominator are raised to the 6th power separately, which is consistent with the properties of exponents:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
4. Option D: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This option simply multiplies [tex]\(\frac{4}{5}\)[/tex] by 6, which is not related to raising the fraction to the 6th power.
Therefore, the correct answer is Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]. This option correctly represents [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Let's evaluate the given options:
1. Option A: [tex]\(\frac{24}{30}\)[/tex]
Simplify [tex]\(\frac{24}{30}\)[/tex]:
[tex]\[
\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}
\][/tex]
This is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
2. Option B: [tex]\(\frac{4^6}{5}\)[/tex]
This expression is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] because the denominator is not raised to the 6th power.
3. Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
This expression represents [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] because both the numerator and the denominator are raised to the 6th power separately, which is consistent with the properties of exponents:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
4. Option D: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This option simply multiplies [tex]\(\frac{4}{5}\)[/tex] by 6, which is not related to raising the fraction to the 6th power.
Therefore, the correct answer is Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]. This option correctly represents [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
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