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Answer :
To determine which option is equal to the fraction [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], let's evaluate each option separately:
1. Original Expression:
The original expression is [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
2. Option A: [tex]\(\frac{4^6}{5^6}\)[/tex]
- When you raise a fraction to a power, both the numerator and the denominator are raised to that power.
- [tex]\(\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}\)[/tex].
- This matches the original expression exactly.
3. Option B: [tex]\(\frac{24}{30}\)[/tex]
- Simplifying [tex]\(\frac{24}{30}\)[/tex] gives [tex]\(\frac{4}{5}\)[/tex], which is not the same as raising to the 6th power.
- Therefore, this does not equal [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
4. Option C: [tex]\(\frac{4^6}{5}\)[/tex]
- This representation does not raise the denominator to the 6th power, so it does not match the original expression.
5. Option D: [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex]
- Multiplying [tex]\(6\)[/tex] by [tex]\(\frac{4}{5}\)[/tex] does not involve an exponent of 6.
- This is different from the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
After comparing each option, the only expression that accurately represents [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is Option A: [tex]\(\frac{4^6}{5^6}\)[/tex].
1. Original Expression:
The original expression is [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
2. Option A: [tex]\(\frac{4^6}{5^6}\)[/tex]
- When you raise a fraction to a power, both the numerator and the denominator are raised to that power.
- [tex]\(\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}\)[/tex].
- This matches the original expression exactly.
3. Option B: [tex]\(\frac{24}{30}\)[/tex]
- Simplifying [tex]\(\frac{24}{30}\)[/tex] gives [tex]\(\frac{4}{5}\)[/tex], which is not the same as raising to the 6th power.
- Therefore, this does not equal [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
4. Option C: [tex]\(\frac{4^6}{5}\)[/tex]
- This representation does not raise the denominator to the 6th power, so it does not match the original expression.
5. Option D: [tex]\(6 \cdot\left(\frac{4}{5}\right)\)[/tex]
- Multiplying [tex]\(6\)[/tex] by [tex]\(\frac{4}{5}\)[/tex] does not involve an exponent of 6.
- This is different from the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
After comparing each option, the only expression that accurately represents [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is Option A: [tex]\(\frac{4^6}{5^6}\)[/tex].
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