Answer :
Sure! Let's solve the problem step by step to find out which option is equal to [tex]\((\frac{4}{5})^6\)[/tex].
1. Simplify the expression [tex]\((\frac{4}{5})^6\)[/tex]:
When you raise a fraction to a power, you apply the exponent to both the numerator and the denominator separately. So, [tex]\((\frac{4}{5})^6\)[/tex] can be rewritten as:
[tex]\[
\frac{4^6}{5^6}
\][/tex]
2. Evaluate each option:
- Option A: [tex]\(\frac{4^6}{5}\)[/tex]
Here, the denominator isn't raised to the power of 6, which is different from our expression.
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex]
This matches the form [tex]\(\frac{4^6}{5^6}\)[/tex], which is exactly what we derived from [tex]\((\frac{4}{5})^6\)[/tex].
- Option C: [tex]\(\frac{24}{30}\)[/tex]
This fraction can be simplified by dividing the numerator and denominator by 6, resulting in [tex]\(\frac{4}{5}\)[/tex], which doesn't match the original expression raised to the sixth power.
- Option D: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This is simpler arithmetic, which results in [tex]\( \frac{24}{5} \)[/tex], and is clearly not equal to [tex]\((\frac{4}{5})^6\)[/tex].
3. Conclusion:
Among all the given choices, only Option B: [tex]\(\frac{4^6}{5^6}\)[/tex] correctly represents [tex]\((\frac{4}{5})^6\)[/tex].
So, the correct answer is Option B: [tex]\(\frac{4^6}{5^6}\)[/tex].
1. Simplify the expression [tex]\((\frac{4}{5})^6\)[/tex]:
When you raise a fraction to a power, you apply the exponent to both the numerator and the denominator separately. So, [tex]\((\frac{4}{5})^6\)[/tex] can be rewritten as:
[tex]\[
\frac{4^6}{5^6}
\][/tex]
2. Evaluate each option:
- Option A: [tex]\(\frac{4^6}{5}\)[/tex]
Here, the denominator isn't raised to the power of 6, which is different from our expression.
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex]
This matches the form [tex]\(\frac{4^6}{5^6}\)[/tex], which is exactly what we derived from [tex]\((\frac{4}{5})^6\)[/tex].
- Option C: [tex]\(\frac{24}{30}\)[/tex]
This fraction can be simplified by dividing the numerator and denominator by 6, resulting in [tex]\(\frac{4}{5}\)[/tex], which doesn't match the original expression raised to the sixth power.
- Option D: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
This is simpler arithmetic, which results in [tex]\( \frac{24}{5} \)[/tex], and is clearly not equal to [tex]\((\frac{4}{5})^6\)[/tex].
3. Conclusion:
Among all the given choices, only Option B: [tex]\(\frac{4^6}{5^6}\)[/tex] correctly represents [tex]\((\frac{4}{5})^6\)[/tex].
So, the correct answer is Option B: [tex]\(\frac{4^6}{5^6}\)[/tex].
