To simplify [tex]\( 256^{\frac{5}{4}} \)[/tex]:
1. Express 256 as a power of 2:
[tex]\[
256 = 2^8
\][/tex]
2. Substitute this into the given expression:
[tex]\[
256^{\frac{5}{4}} = (2^8)^{\frac{5}{4}}
\][/tex]
3. Use the power of a power property, [tex]\( (a^m)^n = a^{m \cdot n} \)[/tex]:
[tex]\[
(2^8)^{\frac{5}{4}} = 2^{8 \cdot \frac{5}{4}}
\][/tex]
4. Perform the multiplication in the exponent:
[tex]\[
8 \cdot \frac{5}{4} = 10
\][/tex]
5. Rewrite the expression with the new exponent:
[tex]\[
2^{10}
\][/tex]
6. Evaluate the expression:
[tex]\[
2^{10} = 1024
\][/tex]
So, the simplified form of [tex]\( 256^{\frac{5}{4}} \)[/tex] is:
[tex]\[
256^{\frac{5}{4}} = 1024
\][/tex]
