To simplify [tex]\( 16^{\frac{5}{4}} \)[/tex], we need to evaluate this expression step-by-step.
1. Understand the base and the exponent:
- The base is [tex]\( 16 \)[/tex].
- The exponent is [tex]\( \frac{5}{4} \)[/tex].
2. Rewrite the base as a power of a simpler number:
- Notice that [tex]\( 16 \)[/tex] can be written as [tex]\( 2^4 \)[/tex] because [tex]\( 2^4 = 16 \)[/tex].
So, we can rewrite the expression as:
[tex]\[
16^{\frac{5}{4}} = (2^4)^{\frac{5}{4}}
\][/tex]
3. Apply the exponentiation property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
- Combine the exponents by multiplying them: [tex]\( 4 \cdot \frac{5}{4} = 5 \)[/tex].
Therefore:
[tex]\[
(2^4)^{\frac{5}{4}} = 2^{4 \cdot \frac{5}{4}} = 2^5
\][/tex]
4. Calculate the final power:
- Simplify [tex]\( 2^5 \)[/tex].
We know:
[tex]\[
2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32
\][/tex]
So, the simplified result of [tex]\( 16^{\frac{5}{4}} \)[/tex] is:
[tex]\[
\boxed{32}
\][/tex]
