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Answer :
To solve this problem, we're trying to find which of the options is equal to the fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Let's look at each option:
Option A: [tex]\(\frac{4^6}{5^6}\)[/tex]
This option represents exactly how [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is mathematically expressed. When you raise a fraction to a power, you raise both the numerator and the denominator to that power. Therefore, this option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Option B: [tex]\(\frac{24}{30}\)[/tex]
To compare this fraction with [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], we need to simplify it. The fraction [tex]\(\frac{24}{30}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex]. However, [tex]\(\frac{4}{5}\)[/tex] is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], because [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means [tex]\(\frac{4}{5}\)[/tex] multiplied by itself six times.
Option C: [tex]\(\frac{4^6}{5}\)[/tex]
This option shows only the numerator raised to the 6th power, while the denominator is not. This is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], where both the numerator and the denominator should be raised to the 6th power.
Option D: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This expression multiplies [tex]\(\frac{4}{5}\)[/tex] by 6, which doesn't involve raising anything to a power. This is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Conclusion:
Only Option A, [tex]\(\frac{4^6}{5^6}\)[/tex], matches the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] as raising both the numerator and the denominator to the 6th power. Thus, Option A is the correct choice.
Let's look at each option:
Option A: [tex]\(\frac{4^6}{5^6}\)[/tex]
This option represents exactly how [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is mathematically expressed. When you raise a fraction to a power, you raise both the numerator and the denominator to that power. Therefore, this option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Option B: [tex]\(\frac{24}{30}\)[/tex]
To compare this fraction with [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], we need to simplify it. The fraction [tex]\(\frac{24}{30}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex]. However, [tex]\(\frac{4}{5}\)[/tex] is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], because [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means [tex]\(\frac{4}{5}\)[/tex] multiplied by itself six times.
Option C: [tex]\(\frac{4^6}{5}\)[/tex]
This option shows only the numerator raised to the 6th power, while the denominator is not. This is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], where both the numerator and the denominator should be raised to the 6th power.
Option D: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This expression multiplies [tex]\(\frac{4}{5}\)[/tex] by 6, which doesn't involve raising anything to a power. This is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Conclusion:
Only Option A, [tex]\(\frac{4^6}{5^6}\)[/tex], matches the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] as raising both the numerator and the denominator to the 6th power. Thus, Option A is the correct choice.
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