Answer :
To solve the problem of determining which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], we'll break it down into steps:
1. Understand the Expression: We have the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] which means we need to raise both the numerator (4) and the denominator (5) to the sixth power.
2. Calculate the Numerator:
- [tex]\(4^6 = 4096\)[/tex] (This involves multiplying 4 by itself six times: [tex]\(4 \times 4 \times 4 \times 4 \times 4 \times 4\)[/tex]).
3. Calculate the Denominator:
- [tex]\(5^6 = 15625\)[/tex] (Again, multiplying 5 by itself six times: [tex]\(5 \times 5 \times 5 \times 5 \times 5 \times 5\)[/tex]).
4. Form the Fraction:
- The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] becomes [tex]\(\frac{4096}{15625}\)[/tex].
5. Match the Fraction to the Options:
- Option A: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex] is different because it doesn’t involve raising 4/5 to the sixth power.
- Option B: [tex]\(\frac{24}{30}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex], which is not the same as [tex]\(\frac{4096}{15625}\)[/tex].
- Option C: [tex]\(\frac{4^6}{5}\)[/tex] does not correctly represent the operation because the denominator should also be raised to the sixth power.
- Option D: [tex]\(\frac{4^6}{5^6}\)[/tex] matches perfectly as it represents the full expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Therefore, the correct choice is Option D: [tex]\(\frac{4^6}{5^6}\)[/tex].
1. Understand the Expression: We have the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] which means we need to raise both the numerator (4) and the denominator (5) to the sixth power.
2. Calculate the Numerator:
- [tex]\(4^6 = 4096\)[/tex] (This involves multiplying 4 by itself six times: [tex]\(4 \times 4 \times 4 \times 4 \times 4 \times 4\)[/tex]).
3. Calculate the Denominator:
- [tex]\(5^6 = 15625\)[/tex] (Again, multiplying 5 by itself six times: [tex]\(5 \times 5 \times 5 \times 5 \times 5 \times 5\)[/tex]).
4. Form the Fraction:
- The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] becomes [tex]\(\frac{4096}{15625}\)[/tex].
5. Match the Fraction to the Options:
- Option A: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex] is different because it doesn’t involve raising 4/5 to the sixth power.
- Option B: [tex]\(\frac{24}{30}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex], which is not the same as [tex]\(\frac{4096}{15625}\)[/tex].
- Option C: [tex]\(\frac{4^6}{5}\)[/tex] does not correctly represent the operation because the denominator should also be raised to the sixth power.
- Option D: [tex]\(\frac{4^6}{5^6}\)[/tex] matches perfectly as it represents the full expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Therefore, the correct choice is Option D: [tex]\(\frac{4^6}{5^6}\)[/tex].
