Answer :
To find which expression is equal to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex], we'll evaluate [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] by raising both the numerator and the denominator separately to the power of 6.
1. Raise the numerator (4) to the power of 6:
[tex]\[
4^6 = 4096
\][/tex]
2. Raise the denominator (5) to the power of 6:
[tex]\[
5^6 = 15625
\][/tex]
Therefore, the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] simplifies to:
[tex]\[
\frac{4^6}{5^6} = \frac{4096}{15625}
\][/tex]
Now, let's compare this expression to the given options:
A. [tex]\(\frac{4^6}{5}\)[/tex] - This does not match since 5 is not raised to the power 6.
B. [tex]\(\frac{24}{30}\)[/tex] - This does not match since neither the numerator nor the denominator matches [tex]\(\frac{4096}{15625}\)[/tex].
C. [tex]\(\frac{4^6}{5^6}\)[/tex] - This is exactly what we calculated.
D. [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex] - This is not raised to the power 6, so it does not match.
The correct expression equivalent to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is option C: [tex]\(\frac{4^6}{5^6}\)[/tex].
1. Raise the numerator (4) to the power of 6:
[tex]\[
4^6 = 4096
\][/tex]
2. Raise the denominator (5) to the power of 6:
[tex]\[
5^6 = 15625
\][/tex]
Therefore, the expression [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] simplifies to:
[tex]\[
\frac{4^6}{5^6} = \frac{4096}{15625}
\][/tex]
Now, let's compare this expression to the given options:
A. [tex]\(\frac{4^6}{5}\)[/tex] - This does not match since 5 is not raised to the power 6.
B. [tex]\(\frac{24}{30}\)[/tex] - This does not match since neither the numerator nor the denominator matches [tex]\(\frac{4096}{15625}\)[/tex].
C. [tex]\(\frac{4^6}{5^6}\)[/tex] - This is exactly what we calculated.
D. [tex]\(6 \cdot \left(\frac{4}{5}\right)\)[/tex] - This is not raised to the power 6, so it does not match.
The correct expression equivalent to [tex]\(\left(\frac{4}{5}\right)^6\)[/tex] is option C: [tex]\(\frac{4^6}{5^6}\)[/tex].
