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Answer :
To find which option is equivalent to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], we need to evaluate each of the given options.
The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] evaluates to approximately 0.262144.
Now, let's evaluate each option:
A. [tex]\(\frac{4^6}{5}\)[/tex]
First, calculate [tex]\(4^6\)[/tex], which is equal to 4096. Then, divide by 5:
[tex]\[
\frac{4096}{5} = 819.2
\][/tex]
This does not match 0.262144.
B. [tex]\(\frac{4^6}{5^6}\)[/tex]
Calculate [tex]\(4^6\)[/tex] which is 4096, and [tex]\(5^6\)[/tex] which is 15625. Divide the two:
[tex]\[
\frac{4096}{15625} \approx 0.262144
\][/tex]
This matches 0.262144.
C. [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex]
First, calculate [tex]\(\frac{4}{5}\)[/tex], which is 0.8. Then multiply by 6:
[tex]\[
6 \times 0.8 = 4.8
\][/tex]
This does not match 0.262144.
D. [tex]\(\frac{24}{30}\)[/tex]
Simplify [tex]\(\frac{24}{30}\)[/tex] to [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[
\frac{4}{5} = 0.8
\][/tex]
This does not match 0.262144.
After evaluating all options, the correct choice is B: [tex]\(\frac{4^6}{5^6}\)[/tex].
The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] evaluates to approximately 0.262144.
Now, let's evaluate each option:
A. [tex]\(\frac{4^6}{5}\)[/tex]
First, calculate [tex]\(4^6\)[/tex], which is equal to 4096. Then, divide by 5:
[tex]\[
\frac{4096}{5} = 819.2
\][/tex]
This does not match 0.262144.
B. [tex]\(\frac{4^6}{5^6}\)[/tex]
Calculate [tex]\(4^6\)[/tex] which is 4096, and [tex]\(5^6\)[/tex] which is 15625. Divide the two:
[tex]\[
\frac{4096}{15625} \approx 0.262144
\][/tex]
This matches 0.262144.
C. [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex]
First, calculate [tex]\(\frac{4}{5}\)[/tex], which is 0.8. Then multiply by 6:
[tex]\[
6 \times 0.8 = 4.8
\][/tex]
This does not match 0.262144.
D. [tex]\(\frac{24}{30}\)[/tex]
Simplify [tex]\(\frac{24}{30}\)[/tex] to [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[
\frac{4}{5} = 0.8
\][/tex]
This does not match 0.262144.
After evaluating all options, the correct choice is B: [tex]\(\frac{4^6}{5^6}\)[/tex].
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