Answer :
To find which option is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], let's analyze each choice:
1. Calculating [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means you multiply the fraction [tex]\(\frac{4}{5}\)[/tex] by itself 6 times. This results in:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
This is because when you raise a fraction to a power, you raise both the numerator and the denominator to that power.
- [tex]\(4^6 = 4096\)[/tex]
- [tex]\(5^6 = 15625\)[/tex]
So, [tex]\(\left(\frac{4}{5}\right)^6 = \frac{4096}{15625}\)[/tex].
2. Now, let's compare this with each option:
A. [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
- This expression is simply multiplying the fraction [tex]\(\frac{4}{5}\)[/tex] by 6, not raising it to the sixth power. It does not match [tex]\(\frac{4^6}{5^6}\)[/tex].
B. [tex]\(\frac{4^6}{5}\)[/tex]
- This would mean only the numerator (4) is raised to the sixth power, leaving the denominator as just 5, which is incorrect compared to [tex]\(\frac{4^6}{5^6}\)[/tex].
C. [tex]\(\frac{4^6}{5^6}\)[/tex]
- This exactly matches our calculated result.
D. [tex]\(\frac{24}{30}\)[/tex]
- This fraction simplifies to [tex]\(\frac{4}{5}\)[/tex] but not raised to any power. It does not match what we are looking for.
Therefore, option C. [tex]\(\frac{4^6}{5^6}\)[/tex] is the correct answer as it matches the original expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
1. Calculating [tex]\(\left(\frac{4}{5}\right)^6\)[/tex]:
The expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] means you multiply the fraction [tex]\(\frac{4}{5}\)[/tex] by itself 6 times. This results in:
[tex]\[
\left(\frac{4}{5}\right)^6 = \frac{4^6}{5^6}
\][/tex]
This is because when you raise a fraction to a power, you raise both the numerator and the denominator to that power.
- [tex]\(4^6 = 4096\)[/tex]
- [tex]\(5^6 = 15625\)[/tex]
So, [tex]\(\left(\frac{4}{5}\right)^6 = \frac{4096}{15625}\)[/tex].
2. Now, let's compare this with each option:
A. [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex]
- This expression is simply multiplying the fraction [tex]\(\frac{4}{5}\)[/tex] by 6, not raising it to the sixth power. It does not match [tex]\(\frac{4^6}{5^6}\)[/tex].
B. [tex]\(\frac{4^6}{5}\)[/tex]
- This would mean only the numerator (4) is raised to the sixth power, leaving the denominator as just 5, which is incorrect compared to [tex]\(\frac{4^6}{5^6}\)[/tex].
C. [tex]\(\frac{4^6}{5^6}\)[/tex]
- This exactly matches our calculated result.
D. [tex]\(\frac{24}{30}\)[/tex]
- This fraction simplifies to [tex]\(\frac{4}{5}\)[/tex] but not raised to any power. It does not match what we are looking for.
Therefore, option C. [tex]\(\frac{4^6}{5^6}\)[/tex] is the correct answer as it matches the original expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex].
