To simplify the expression [tex]\(\left(\frac{2 a^4}{5}\right)^4\)[/tex], follow these steps:
1. Start with the expression:
[tex]\[
\left(\frac{2 a^4}{5}\right)^4
\][/tex]
2. Apply the power to both the numerator and the denominator:
[tex]\[
\left(\frac{2 a^4}{5}\right)^4 = \frac{(2 a^4)^4}{5^4}
\][/tex]
3. Calculate the numerator:
[tex]\[
(2 a^4)^4
\][/tex]
Here, raise both 2 and [tex]\(a^4\)[/tex] to the 4th power:
[tex]\[
(2 a^4)^4 = 2^4 \cdot (a^4)^4
\][/tex]
4. Simplify the powers:
[tex]\[
2^4 = 16
\][/tex]
[tex]\[
(a^4)^4 = a^{4 \cdot 4} = a^{16}
\][/tex]
So, the numerator becomes:
[tex]\[
2^4 \cdot (a^4)^4 = 16 a^{16}
\][/tex]
5. Calculate the denominator:
[tex]\[
5^4 = 625
\][/tex]
6. Combine the numerator and denominator:
[tex]\[
\frac{16 a^{16}}{625}
\][/tex]
Thus, the simplified form of [tex]\(\left(\frac{2 a^4}{5}\right)^4\)[/tex] is:
[tex]\[
\boxed{\frac{16 a^{16}}{625}}
\][/tex]
