Answer :
To identify which choice is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], we need to analyze each option:
1. Calculate [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
- [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means multiplying the fraction [tex]\(\frac{4}{5}\)[/tex] by itself six times. numerically, this equals approximately 0.262144.
2. Evaluate each choice:
- Choice A: [tex]\(6 \cdot \binom{4}{5}\)[/tex]
- [tex]\(\binom{4}{5}\)[/tex] is a combination calculation, which means choosing 5 items from 4. Since it's impossible to choose more items than available, the value is 0. Hence, [tex]\(6 \cdot 0 = 0\)[/tex].
- Choice B: [tex]\(\frac{24}{30}\)[/tex]
- Simplifying [tex]\(\frac{24}{30}\)[/tex] gives [tex]\(\frac{4}{5}\)[/tex], which is approximately 0.8.
- Choice C: [tex]\(\frac{4^6}{5}\)[/tex]
- Calculate [tex]\(4^6\)[/tex], which is 4096. Then divide by 5: [tex]\(\frac{4096}{5} = 819.2\)[/tex].
- Choice D: [tex]\(\frac{4^6}{5^6}\)[/tex]
- Calculate [tex]\(4^6 = 4096\)[/tex] and [tex]\(5^6 = 15625\)[/tex].
- Then compute [tex]\(\frac{4096}{15625}\)[/tex], which is approximately 0.262144.
3. Conclusion:
- Comparing the values, we find that Choice D, [tex]\(\frac{4^6}{5^6}\)[/tex], equals approximately 0.262144, which matches [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Therefore, the correct choice is D: [tex]\(\frac{4^6}{5^6}\)[/tex].
1. Calculate [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
- [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means multiplying the fraction [tex]\(\frac{4}{5}\)[/tex] by itself six times. numerically, this equals approximately 0.262144.
2. Evaluate each choice:
- Choice A: [tex]\(6 \cdot \binom{4}{5}\)[/tex]
- [tex]\(\binom{4}{5}\)[/tex] is a combination calculation, which means choosing 5 items from 4. Since it's impossible to choose more items than available, the value is 0. Hence, [tex]\(6 \cdot 0 = 0\)[/tex].
- Choice B: [tex]\(\frac{24}{30}\)[/tex]
- Simplifying [tex]\(\frac{24}{30}\)[/tex] gives [tex]\(\frac{4}{5}\)[/tex], which is approximately 0.8.
- Choice C: [tex]\(\frac{4^6}{5}\)[/tex]
- Calculate [tex]\(4^6\)[/tex], which is 4096. Then divide by 5: [tex]\(\frac{4096}{5} = 819.2\)[/tex].
- Choice D: [tex]\(\frac{4^6}{5^6}\)[/tex]
- Calculate [tex]\(4^6 = 4096\)[/tex] and [tex]\(5^6 = 15625\)[/tex].
- Then compute [tex]\(\frac{4096}{15625}\)[/tex], which is approximately 0.262144.
3. Conclusion:
- Comparing the values, we find that Choice D, [tex]\(\frac{4^6}{5^6}\)[/tex], equals approximately 0.262144, which matches [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Therefore, the correct choice is D: [tex]\(\frac{4^6}{5^6}\)[/tex].
