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Answer :
To determine which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], let's examine the choices provided:
We have the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], which means raising both the numerator (4) and the denominator (5) to the 6th power.
Let's break it down:
1. [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is calculated as [tex]\(\frac{4^6}{5^6}\)[/tex].
Now, check each option to find an equivalent expression:
- Option A: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This expression represents multiplying [tex]\(\frac{4}{5}\)[/tex] by 6, which is not the same as raising [tex]\(\frac{4}{5}\)[/tex] to the 6th power.
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex]
This option correctly represents the process of raising both the numerator and denominator to the 6th power, which matches our original expression.
- Option C: [tex]\(\frac{24}{30}\)[/tex]
Although [tex]\(\frac{24}{30}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex], it doesn't represent raising that fraction to the 6th power.
- Option D: [tex]\(\frac{4^6}{5}\)[/tex]
This expression correctly raises 4 to the 6th power but does not raise 5 to the 6th power, making it incorrect.
Therefore, the correct option that equals [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is Option B: [tex]\(\frac{4^6}{5^6}\)[/tex].
We have the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], which means raising both the numerator (4) and the denominator (5) to the 6th power.
Let's break it down:
1. [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is calculated as [tex]\(\frac{4^6}{5^6}\)[/tex].
Now, check each option to find an equivalent expression:
- Option A: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This expression represents multiplying [tex]\(\frac{4}{5}\)[/tex] by 6, which is not the same as raising [tex]\(\frac{4}{5}\)[/tex] to the 6th power.
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex]
This option correctly represents the process of raising both the numerator and denominator to the 6th power, which matches our original expression.
- Option C: [tex]\(\frac{24}{30}\)[/tex]
Although [tex]\(\frac{24}{30}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex], it doesn't represent raising that fraction to the 6th power.
- Option D: [tex]\(\frac{4^6}{5}\)[/tex]
This expression correctly raises 4 to the 6th power but does not raise 5 to the 6th power, making it incorrect.
Therefore, the correct option that equals [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is Option B: [tex]\(\frac{4^6}{5^6}\)[/tex].
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