Answer :
To solve this question, we need to determine which option is equal to the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
Let's break it down:
1. Understanding the Expression:
- The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means we are raising the fraction [tex]\(\frac{4}{5}\)[/tex] to the sixth power.
- This can be written as [tex]\(\frac{4^6}{5^6}\)[/tex] because when you raise a fraction to a power, you raise both the numerator and the denominator to that power.
2. Evaluating the Options:
- Option A: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
- This option is simply multiplying the fraction [tex]\(\frac{4}{5}\)[/tex] by 6, which is not the same as raising it to the sixth power.
- Option B: [tex]\(\frac{4^6}{5}\)[/tex]
- This option only raises the numerator 4 to the sixth power, leaving the denominator as 5, which is incorrect.
- Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
- This matches our expression [tex]\(\frac{4^6}{5^6}\)[/tex] since it correctly raises both the numerator and denominator to the sixth power.
- Option D: [tex]\(\frac{24}{30}\)[/tex]
- This fraction is a simplification of [tex]\(\frac{4}{5}\)[/tex], not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
3. Conclusion:
- The correct choice that matches the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is Option C: [tex]\(\frac{4^6}{5^6}\)[/tex].
Therefore, the answer to the question is option C.
Let's break it down:
1. Understanding the Expression:
- The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means we are raising the fraction [tex]\(\frac{4}{5}\)[/tex] to the sixth power.
- This can be written as [tex]\(\frac{4^6}{5^6}\)[/tex] because when you raise a fraction to a power, you raise both the numerator and the denominator to that power.
2. Evaluating the Options:
- Option A: [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
- This option is simply multiplying the fraction [tex]\(\frac{4}{5}\)[/tex] by 6, which is not the same as raising it to the sixth power.
- Option B: [tex]\(\frac{4^6}{5}\)[/tex]
- This option only raises the numerator 4 to the sixth power, leaving the denominator as 5, which is incorrect.
- Option C: [tex]\(\frac{4^6}{5^6}\)[/tex]
- This matches our expression [tex]\(\frac{4^6}{5^6}\)[/tex] since it correctly raises both the numerator and denominator to the sixth power.
- Option D: [tex]\(\frac{24}{30}\)[/tex]
- This fraction is a simplification of [tex]\(\frac{4}{5}\)[/tex], not [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
3. Conclusion:
- The correct choice that matches the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is Option C: [tex]\(\frac{4^6}{5^6}\)[/tex].
Therefore, the answer to the question is option C.
