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Answer :
To solve the problem of finding which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], we need to evaluate this expression:
1. Understand the Expression:
- [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means raising the fraction [tex]\(\frac{4}{5}\)[/tex] to the 6th power.
2. Expand the Expression:
- To raise a fraction to a power, raise both the numerator and the denominator to that power.
- This becomes [tex]\(\frac{4^6}{5^6}\)[/tex].
3. Evaluate Each Option:
- Option A: [tex]\(\frac{24}{30}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex], which is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option B: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex] is a linear multiplication, not an exponentiation, so it does not match [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option C: [tex]\(\frac{4^6}{5^6}\)[/tex] matches exactly with how we expanded [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option D: [tex]\(\frac{4^6}{5}\)[/tex] only raises the numerator 4 to the 6th power, but not the denominator correctly, so it's not correct.
4. Conclusion:
- The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is indeed equal to [tex]\(\frac{4^6}{5^6}\)[/tex].
- Therefore, the correct answer is Option C.
1. Understand the Expression:
- [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means raising the fraction [tex]\(\frac{4}{5}\)[/tex] to the 6th power.
2. Expand the Expression:
- To raise a fraction to a power, raise both the numerator and the denominator to that power.
- This becomes [tex]\(\frac{4^6}{5^6}\)[/tex].
3. Evaluate Each Option:
- Option A: [tex]\(\frac{24}{30}\)[/tex] simplifies to [tex]\(\frac{4}{5}\)[/tex], which is not equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option B: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex] is a linear multiplication, not an exponentiation, so it does not match [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option C: [tex]\(\frac{4^6}{5^6}\)[/tex] matches exactly with how we expanded [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
- Option D: [tex]\(\frac{4^6}{5}\)[/tex] only raises the numerator 4 to the 6th power, but not the denominator correctly, so it's not correct.
4. Conclusion:
- The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is indeed equal to [tex]\(\frac{4^6}{5^6}\)[/tex].
- Therefore, the correct answer is Option C.
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