Answer :
To solve the problem of finding which expression is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], we can break it down step by step:
1. Understand the Expression:
- The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means we raise both the numerator and the denominator to the power of 6.
- This becomes [tex]\(\frac{4^6}{5^6}\)[/tex].
2. Explore Each Option:
- Option A: [tex]\(\frac{4^6}{5}\)[/tex]
- This expression raises only the numerator [tex]\(4^6\)[/tex] but does not raise the denominator 5 to the same power. It is not the correct match because both numerator and denominator need to be raised to the power of 6.
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex]
- This option correctly raises both the 4 and the 5 to the power of 6, which matches [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option C: [tex]\(\frac{24}{30}\)[/tex]
- This fraction simplifies to [tex]\(\frac{4}{5}\)[/tex] rather than carrying the power of 6. It doesn't match our original expression when raised to any power.
- Option D: [tex]\(6 \bullet \left(\frac{4}{5}\right)\)[/tex]
- This option multiplies [tex]\(\frac{4}{5}\)[/tex] by 6 but does not raise it to the power of 6. This also doesn't match the original expression.
3. Conclusion:
- After analyzing all the options, the correct choice is B: [tex]\(\frac{4^6}{5^6}\)[/tex]. It correctly represents [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] by applying the exponent to both the numerator and the denominator.
Therefore, the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is equal to [tex]\(\frac{4^6}{5^6}\)[/tex].
1. Understand the Expression:
- The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means we raise both the numerator and the denominator to the power of 6.
- This becomes [tex]\(\frac{4^6}{5^6}\)[/tex].
2. Explore Each Option:
- Option A: [tex]\(\frac{4^6}{5}\)[/tex]
- This expression raises only the numerator [tex]\(4^6\)[/tex] but does not raise the denominator 5 to the same power. It is not the correct match because both numerator and denominator need to be raised to the power of 6.
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex]
- This option correctly raises both the 4 and the 5 to the power of 6, which matches [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option C: [tex]\(\frac{24}{30}\)[/tex]
- This fraction simplifies to [tex]\(\frac{4}{5}\)[/tex] rather than carrying the power of 6. It doesn't match our original expression when raised to any power.
- Option D: [tex]\(6 \bullet \left(\frac{4}{5}\right)\)[/tex]
- This option multiplies [tex]\(\frac{4}{5}\)[/tex] by 6 but does not raise it to the power of 6. This also doesn't match the original expression.
3. Conclusion:
- After analyzing all the options, the correct choice is B: [tex]\(\frac{4^6}{5^6}\)[/tex]. It correctly represents [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] by applying the exponent to both the numerator and the denominator.
Therefore, the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is equal to [tex]\(\frac{4^6}{5^6}\)[/tex].
