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Answer :
To solve the problem of which expression is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], let's take a look at each option provided:
The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
- This means raising the fraction [tex]\(\frac{4}{5}\)[/tex] to the sixth power, which involves multiplying [tex]\(\frac{4}{5}\)[/tex] by itself six times.
Evaluating each option:
1. Option A: [tex]\(\frac{4^6}{5}\)[/tex]
[tex]\(\frac{4^6}{5}\)[/tex] means calculating [tex]\(4^6\)[/tex] and then dividing it by 5.
- If you compute [tex]\(4^6\)[/tex], it's 4 multiplied by itself six times, which equals 4096.
- Therefore, [tex]\(\frac{4^6}{5} = \frac{4096}{5} = 819.2\)[/tex].
2. Option B: [tex]\(6 \cdot \binom{4}{5}\)[/tex]
- The expression [tex]\(\binom{4}{5}\)[/tex] is a binomial coefficient, representing the number of ways to choose 5 items from 4, which is impossible, so equals 0.
- Thus, [tex]\(6 \cdot 0 = 0\)[/tex].
3. Option C: [tex]\(\frac{24}{30}\)[/tex]
- Simplifying [tex]\(\frac{24}{30}\)[/tex] gives [tex]\(\frac{4}{5}\)[/tex]. However, this is not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
4. Option D: [tex]\(\frac{4}{5}_{5^6}\)[/tex]
- This notation is unusual and likely incorrect. The use of subscripts in fractions is not valid, making this option unclear.
Given the evaluations, none of the options match the value of [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], which is approximately 0.262144. Therefore, none of the given choices are equal to the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
- This means raising the fraction [tex]\(\frac{4}{5}\)[/tex] to the sixth power, which involves multiplying [tex]\(\frac{4}{5}\)[/tex] by itself six times.
Evaluating each option:
1. Option A: [tex]\(\frac{4^6}{5}\)[/tex]
[tex]\(\frac{4^6}{5}\)[/tex] means calculating [tex]\(4^6\)[/tex] and then dividing it by 5.
- If you compute [tex]\(4^6\)[/tex], it's 4 multiplied by itself six times, which equals 4096.
- Therefore, [tex]\(\frac{4^6}{5} = \frac{4096}{5} = 819.2\)[/tex].
2. Option B: [tex]\(6 \cdot \binom{4}{5}\)[/tex]
- The expression [tex]\(\binom{4}{5}\)[/tex] is a binomial coefficient, representing the number of ways to choose 5 items from 4, which is impossible, so equals 0.
- Thus, [tex]\(6 \cdot 0 = 0\)[/tex].
3. Option C: [tex]\(\frac{24}{30}\)[/tex]
- Simplifying [tex]\(\frac{24}{30}\)[/tex] gives [tex]\(\frac{4}{5}\)[/tex]. However, this is not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
4. Option D: [tex]\(\frac{4}{5}_{5^6}\)[/tex]
- This notation is unusual and likely incorrect. The use of subscripts in fractions is not valid, making this option unclear.
Given the evaluations, none of the options match the value of [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], which is approximately 0.262144. Therefore, none of the given choices are equal to the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
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