Answer :
Let's solve the problem to find which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
1. Calculate [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
The fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] evaluates approximately to 0.262144.
2. Check each option:
Option A: [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex]
[tex]\(6 \times \frac{4}{5} = \frac{24}{5} = 4.8\)[/tex]
This is not approximately 0.262144, so Option A is incorrect.
Option B: [tex]\(\frac{4^6}{5}\)[/tex]
First compute [tex]\(4^6 = 4096\)[/tex], then divide by 5:
[tex]\(\frac{4096}{5} = 819.2\)[/tex]
This is not approximately 0.262144, so Option B is incorrect.
Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
Again, [tex]\(4^6 = 4096\)[/tex] and [tex]\(5^6 = 15625\)[/tex]:
[tex]\(\frac{4096}{15625} \approx 0.262144\)[/tex]
This is approximately equal to 0.262144, making Option C correct.
Option D: [tex]\(\frac{24}{30}\)[/tex]
Simplifying this fraction: [tex]\(\frac{24}{30} = \frac{4}{5} = 0.8\)[/tex]
This is not approximately 0.262144, so Option D is incorrect.
3. Conclusion:
The option that is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is Option C: [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] evaluates approximately to 0.262144.
2. Check each option:
Option A: [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex]
[tex]\(6 \times \frac{4}{5} = \frac{24}{5} = 4.8\)[/tex]
This is not approximately 0.262144, so Option A is incorrect.
Option B: [tex]\(\frac{4^6}{5}\)[/tex]
First compute [tex]\(4^6 = 4096\)[/tex], then divide by 5:
[tex]\(\frac{4096}{5} = 819.2\)[/tex]
This is not approximately 0.262144, so Option B is incorrect.
Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
Again, [tex]\(4^6 = 4096\)[/tex] and [tex]\(5^6 = 15625\)[/tex]:
[tex]\(\frac{4096}{15625} \approx 0.262144\)[/tex]
This is approximately equal to 0.262144, making Option C correct.
Option D: [tex]\(\frac{24}{30}\)[/tex]
Simplifying this fraction: [tex]\(\frac{24}{30} = \frac{4}{5} = 0.8\)[/tex]
This is not approximately 0.262144, so Option D is incorrect.
3. Conclusion:
The option that is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is Option C: [tex]\(\frac{4^6}{5^6}\)[/tex].
