Answer :
To solve the problem of determining which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], we need to simplify [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] and compare it with the choices provided.
Here are the steps:
1. Calculate [tex]\((\frac{4}{5})^6\)[/tex]:
- This expression means you need to raise both the numerator 4 and the denominator 5 to the power of 6.
- This results in [tex]\(\frac{4^6}{5^6}\)[/tex].
2. Compute [tex]\(4^6\)[/tex] and [tex]\(5^6\)[/tex]:
- [tex]\(4^6 = 4096\)[/tex], as [tex]\(4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4096\)[/tex].
- [tex]\(5^6 = 15625\)[/tex], because [tex]\(5 \times 5 \times 5 \times 5 \times 5 \times 5 = 15625\)[/tex].
3. The fraction becomes:
- [tex]\(\frac{4096}{15625}\)[/tex].
4. Compare this with the answer options:
- Option A: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex] does not involve exponentiation, so it doesn't match.
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex] matches exactly with our calculated fraction, [tex]\(\frac{4096}{15625}\)[/tex].
- Option C: [tex]\(\frac{24}{30}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex], which is not the sixth power and doesn't match [tex]\(\frac{4096}{15625}\)[/tex].
- Option D: [tex]\(\frac{4^6}{5}\)[/tex] is only [tex]\(\frac{4096}{5}\)[/tex] and doesn't match the denominator [tex]\(5^6\)[/tex].
Thus, the correct answer is Option B, [tex]\(\frac{4^6}{5^6}\)[/tex].
Here are the steps:
1. Calculate [tex]\((\frac{4}{5})^6\)[/tex]:
- This expression means you need to raise both the numerator 4 and the denominator 5 to the power of 6.
- This results in [tex]\(\frac{4^6}{5^6}\)[/tex].
2. Compute [tex]\(4^6\)[/tex] and [tex]\(5^6\)[/tex]:
- [tex]\(4^6 = 4096\)[/tex], as [tex]\(4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4096\)[/tex].
- [tex]\(5^6 = 15625\)[/tex], because [tex]\(5 \times 5 \times 5 \times 5 \times 5 \times 5 = 15625\)[/tex].
3. The fraction becomes:
- [tex]\(\frac{4096}{15625}\)[/tex].
4. Compare this with the answer options:
- Option A: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex] does not involve exponentiation, so it doesn't match.
- Option B: [tex]\(\frac{4^6}{5^6}\)[/tex] matches exactly with our calculated fraction, [tex]\(\frac{4096}{15625}\)[/tex].
- Option C: [tex]\(\frac{24}{30}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex], which is not the sixth power and doesn't match [tex]\(\frac{4096}{15625}\)[/tex].
- Option D: [tex]\(\frac{4^6}{5}\)[/tex] is only [tex]\(\frac{4096}{5}\)[/tex] and doesn't match the denominator [tex]\(5^6\)[/tex].
Thus, the correct answer is Option B, [tex]\(\frac{4^6}{5^6}\)[/tex].
