To solve for [tex]\( f\left( \frac{5}{2} \right) \)[/tex] when given the function [tex]\( f(x) = 2^x \)[/tex], follow these steps:
1. Substitute [tex]\( x \)[/tex] with [tex]\( \frac{5}{2} \)[/tex] in the function:
[tex]\[
f\left( \frac{5}{2} \right) = 2^{\frac{5}{2}}
\][/tex]
2. Express the exponent [tex]\( \frac{5}{2} \)[/tex] in a more convenient form:
[tex]\[
2^{\frac{5}{2}} = \left(2^2\right)^{\frac{5}{2 \cdot 2}} = \left(2^2\right)^{\frac{5}{4}}
\][/tex]
3. Simplify the base with the exponent 2:
[tex]\[
2^2 = 4
\][/tex]
So, we rewrite the expression:
[tex]\[
\left(2^2\right)^{\frac{5}{4}} = 4^{\frac{5}{4}}
\][/tex]
4. Find the fourth root of 4 and raise it to the power 5:
[tex]\[
4^{\frac{5}{4}} = \left( 2^2 \right)^{\frac{5}{4}} = ( 2^2 )^{\frac{1}{4} \cdot 5} = 2^{\frac{5 \cdot 2}{4}} = 2^{\frac{5}{2}}
\][/tex]
To express this in terms of radicals:
[tex]\[
2^{\frac{5}{2}} = (2^{\frac{1}{2}})^{5} = (\sqrt{2})^5 = \sqrt{2^5}
\][/tex]
5. Calculate the exact value using radicals:
[tex]\[
2^5 = 32, \quad \therefore \sqrt{32} = 4\sqrt{2}
\][/tex]
Thus, the exact answer is:
[tex]\[
f\left( \frac{5}{2} \right) = 4 \sqrt{2}
\][/tex]
