Answer :
Let's solve the problem step-by-step to determine which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
1. Understand the Expression:
- We are given [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- This means we need to multiply [tex]\(\frac{4}{5}\)[/tex] by itself six times.
2. Evaluate Each Option:
- Option A: [tex]\(\frac{4^6}{5^6}\)[/tex]
- This option uses the same base as the original fraction, raised to the same power. This matches the form of [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option B: [tex]\(\frac{4^6}{5}\)[/tex]
- Here, only [tex]\(4\)[/tex] is raised to the power of 6, but [tex]\(5\)[/tex] is not. This doesn't match our expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option C: [tex]\(\frac{24}{30}\)[/tex]
- Simplifying [tex]\(\frac{24}{30}\)[/tex] gives [tex]\(\frac{4}{5}\)[/tex], which is not in the exponentiated form [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option D: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
- This option multiplies [tex]\(\frac{4}{5}\)[/tex] by 6, instead of raising it to the 6th power, so it is not the correct form.
3. Conclusion:
- Option A, [tex]\(\frac{4^6}{5^6}\)[/tex], is the equivalent expression because it correctly represents the fraction [tex]\(\left(\frac{4}{5}\right)\)[/tex] raised to the 6th power.
Therefore, the correct answer is:
A. [tex]\(\frac{4^6}{5^6}\)[/tex]
1. Understand the Expression:
- We are given [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- This means we need to multiply [tex]\(\frac{4}{5}\)[/tex] by itself six times.
2. Evaluate Each Option:
- Option A: [tex]\(\frac{4^6}{5^6}\)[/tex]
- This option uses the same base as the original fraction, raised to the same power. This matches the form of [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option B: [tex]\(\frac{4^6}{5}\)[/tex]
- Here, only [tex]\(4\)[/tex] is raised to the power of 6, but [tex]\(5\)[/tex] is not. This doesn't match our expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option C: [tex]\(\frac{24}{30}\)[/tex]
- Simplifying [tex]\(\frac{24}{30}\)[/tex] gives [tex]\(\frac{4}{5}\)[/tex], which is not in the exponentiated form [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option D: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
- This option multiplies [tex]\(\frac{4}{5}\)[/tex] by 6, instead of raising it to the 6th power, so it is not the correct form.
3. Conclusion:
- Option A, [tex]\(\frac{4^6}{5^6}\)[/tex], is the equivalent expression because it correctly represents the fraction [tex]\(\left(\frac{4}{5}\right)\)[/tex] raised to the 6th power.
Therefore, the correct answer is:
A. [tex]\(\frac{4^6}{5^6}\)[/tex]
