Answer :
Let's solve the problem by simplifying the expression [tex]\((\frac{4}{5})^6\)[/tex].
1. Write the fraction in the form of a power:
[tex]\[
\left(\frac{4}{5}\right)^6
\][/tex]
2. Apply the power to both the numerator and the denominator:
- The numerator becomes [tex]\(4^6\)[/tex].
- The denominator becomes [tex]\(5^6\)[/tex].
3. Calculate the powers:
- [tex]\(4^6 = 4096\)[/tex]
- [tex]\(5^6 = 15625\)[/tex]
4. Combine the results to form the fraction:
[tex]\[
\frac{4^6}{5^6} = \frac{4096}{15625}
\][/tex]
Now, let's match this result with the options given:
- A. [tex]\(\frac{24}{30}\)[/tex] is not equal to [tex]\(\frac{4096}{15625}\)[/tex].
- B. [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex] yields a different result entirely.
- C. The notation [tex]\(\frac{4}{5}_{5^6}\)[/tex] is unconventional and not equal to [tex]\(\frac{4096}{15625}\)[/tex].
- D. [tex]\(\frac{4^6}{5}\)[/tex] is [tex]\(\frac{4096}{5}\)[/tex], which is not equal to [tex]\(\frac{4096}{15625}\)[/tex].
None of the options provided match the fraction [tex]\(\frac{4096}{15625}\)[/tex], so it seems there may have been an issue with the listed choices. However, this step-by-step solution tells us that the correct response directly from the calculations should match [tex]\(\frac{4096}{15625}\)[/tex] for the expression [tex]\((\frac{4}{5})^6\)[/tex].
1. Write the fraction in the form of a power:
[tex]\[
\left(\frac{4}{5}\right)^6
\][/tex]
2. Apply the power to both the numerator and the denominator:
- The numerator becomes [tex]\(4^6\)[/tex].
- The denominator becomes [tex]\(5^6\)[/tex].
3. Calculate the powers:
- [tex]\(4^6 = 4096\)[/tex]
- [tex]\(5^6 = 15625\)[/tex]
4. Combine the results to form the fraction:
[tex]\[
\frac{4^6}{5^6} = \frac{4096}{15625}
\][/tex]
Now, let's match this result with the options given:
- A. [tex]\(\frac{24}{30}\)[/tex] is not equal to [tex]\(\frac{4096}{15625}\)[/tex].
- B. [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex] yields a different result entirely.
- C. The notation [tex]\(\frac{4}{5}_{5^6}\)[/tex] is unconventional and not equal to [tex]\(\frac{4096}{15625}\)[/tex].
- D. [tex]\(\frac{4^6}{5}\)[/tex] is [tex]\(\frac{4096}{5}\)[/tex], which is not equal to [tex]\(\frac{4096}{15625}\)[/tex].
None of the options provided match the fraction [tex]\(\frac{4096}{15625}\)[/tex], so it seems there may have been an issue with the listed choices. However, this step-by-step solution tells us that the correct response directly from the calculations should match [tex]\(\frac{4096}{15625}\)[/tex] for the expression [tex]\((\frac{4}{5})^6\)[/tex].
