Answer :
To solve the problem, we need to determine which option is equal to the expression [tex]\((\frac{4}{5})^6\)[/tex].
Let's go through each option one by one:
Option A: [tex]\(\frac{24}{30}\)[/tex]
- Simplify [tex]\(\frac{24}{30}\)[/tex]:
- Both 24 and 30 are divisible by 6.
- Hence, [tex]\(\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}\)[/tex].
This value is [tex]\(\frac{4}{5}\)[/tex], not [tex]\((\frac{4}{5})^6\)[/tex].
Option B: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
- Calculate this option:
- [tex]\(6 \cdot \frac{4}{5} = \frac{6 \times 4}{5} = \frac{24}{5}\)[/tex].
This result is [tex]\(\frac{24}{5}\)[/tex], not [tex]\((\frac{4}{5})^6\)[/tex].
Option C: [tex]\(\frac{4^6}{5}\)[/tex]
- Evaluate this option:
- [tex]\(4^6\)[/tex] means 4 raised to the power of 6.
- [tex]\(\frac{4^6}{5}\)[/tex] means dividing this value by 5.
This fraction is essentially [tex]\(\frac{4096}{5}\)[/tex], but we're looking for [tex]\((\frac{4}{5})^6\)[/tex], which is much smaller.
Option D: [tex]\(\frac{4^6}{5^6}\)[/tex]
- Calculate this option:
- This means [tex]\(4^6\)[/tex] divided by [tex]\(5^6\)[/tex].
- This setup matches [tex]\((\frac{4}{5})^6\)[/tex] because it represents [tex]\(\frac{4^6}{5^6}\)[/tex].
Therefore, Option D [tex]\(\frac{4^6}{5^6}\)[/tex] aligns with [tex]\((\frac{4}{5})^6\)[/tex].
So, the correct answer is Option D: [tex]\(\frac{4^6}{5^6}\)[/tex].
Let's go through each option one by one:
Option A: [tex]\(\frac{24}{30}\)[/tex]
- Simplify [tex]\(\frac{24}{30}\)[/tex]:
- Both 24 and 30 are divisible by 6.
- Hence, [tex]\(\frac{24}{30} = \frac{24 \div 6}{30 \div 6} = \frac{4}{5}\)[/tex].
This value is [tex]\(\frac{4}{5}\)[/tex], not [tex]\((\frac{4}{5})^6\)[/tex].
Option B: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
- Calculate this option:
- [tex]\(6 \cdot \frac{4}{5} = \frac{6 \times 4}{5} = \frac{24}{5}\)[/tex].
This result is [tex]\(\frac{24}{5}\)[/tex], not [tex]\((\frac{4}{5})^6\)[/tex].
Option C: [tex]\(\frac{4^6}{5}\)[/tex]
- Evaluate this option:
- [tex]\(4^6\)[/tex] means 4 raised to the power of 6.
- [tex]\(\frac{4^6}{5}\)[/tex] means dividing this value by 5.
This fraction is essentially [tex]\(\frac{4096}{5}\)[/tex], but we're looking for [tex]\((\frac{4}{5})^6\)[/tex], which is much smaller.
Option D: [tex]\(\frac{4^6}{5^6}\)[/tex]
- Calculate this option:
- This means [tex]\(4^6\)[/tex] divided by [tex]\(5^6\)[/tex].
- This setup matches [tex]\((\frac{4}{5})^6\)[/tex] because it represents [tex]\(\frac{4^6}{5^6}\)[/tex].
Therefore, Option D [tex]\(\frac{4^6}{5^6}\)[/tex] aligns with [tex]\((\frac{4}{5})^6\)[/tex].
So, the correct answer is Option D: [tex]\(\frac{4^6}{5^6}\)[/tex].
