Answer :
To determine which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], let's evaluate each choice and compare it to the given fraction:
1. Calculate [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
The fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] represents the number when you multiply [tex]\(\frac{4}{5}\)[/tex] by itself six times.
2. Evaluate the answer choices:
- Option A: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
- This option multiplies [tex]\(6\)[/tex] by [tex]\(\frac{4}{5}\)[/tex]. It's not raising [tex]\(\frac{4}{5}\)[/tex] to the sixth power.
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex]
- This option is formed by calculating [tex]\(4^6\)[/tex] divided by [tex]\(5^6\)[/tex]. This matches the calculation of [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], as you raise both the numerator and denominator to the sixth power.
- Option C: [tex]\(\frac{24}{30}\)[/tex]
- This fraction simplifies to [tex]\(\frac{4}{5}\)[/tex], which is not equivalent to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option D: [tex]\(\frac{4^6}{5}\)[/tex]
- This option only raises the numerator [tex]\(4^6\)[/tex] but keeps the denominator as [tex]\(5\)[/tex], rather than raising it to the sixth power.
3. Conclusion:
The fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] simplifies to the same expression as [tex]\(\frac{4^6}{5^6}\)[/tex]. Therefore, the correct answer is Option B: [tex]\(\frac{4^6}{5^6}\)[/tex].
1. Calculate [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
The fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] represents the number when you multiply [tex]\(\frac{4}{5}\)[/tex] by itself six times.
2. Evaluate the answer choices:
- Option A: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
- This option multiplies [tex]\(6\)[/tex] by [tex]\(\frac{4}{5}\)[/tex]. It's not raising [tex]\(\frac{4}{5}\)[/tex] to the sixth power.
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex]
- This option is formed by calculating [tex]\(4^6\)[/tex] divided by [tex]\(5^6\)[/tex]. This matches the calculation of [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], as you raise both the numerator and denominator to the sixth power.
- Option C: [tex]\(\frac{24}{30}\)[/tex]
- This fraction simplifies to [tex]\(\frac{4}{5}\)[/tex], which is not equivalent to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option D: [tex]\(\frac{4^6}{5}\)[/tex]
- This option only raises the numerator [tex]\(4^6\)[/tex] but keeps the denominator as [tex]\(5\)[/tex], rather than raising it to the sixth power.
3. Conclusion:
The fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] simplifies to the same expression as [tex]\(\frac{4^6}{5^6}\)[/tex]. Therefore, the correct answer is Option B: [tex]\(\frac{4^6}{5^6}\)[/tex].
