To solve the expression [tex]\( 16^{5 / 2} \)[/tex], let's break down the steps in detail:
1. First, we need to understand that [tex]\( 16^{5 / 2} \)[/tex] can be rewritten in radical form. The general rule for rewriting an expression with a fractional exponent [tex]\( a^{b/c} \)[/tex] is:
[tex]\[
a^{b/c} = (a^{1/c})^b
\][/tex]
2. Apply this rule to the given expression. Here, [tex]\( a = 16 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = 2 \)[/tex]:
[tex]\[
16^{5 / 2} = (16^{1/2})^5
\][/tex]
3. Now, [tex]\( 16^{1/2} \)[/tex] is the square root of 16. The square root of 16 is 4:
[tex]\[
16^{1/2} = \sqrt{16} = 4
\][/tex]
4. Substitute [tex]\(\sqrt{16}\)[/tex] with 4 in the expression:
[tex]\[
(16^{1/2})^5 = 4^5
\][/tex]
5. Now, calculate [tex]\( 4^5 \)[/tex]:
[tex]\[
4^5 = 4 \times 4 \times 4 \times 4 \times 4 = 1024
\][/tex]
So, in radical form, the expression [tex]\( 16^{5 / 2} \)[/tex] becomes [tex]\( 4^5 \)[/tex], and its evaluated numerical value is 1024.
Therefore, the correct choice is:
A. [tex]\( 16^{5 / 2} = 4^5 \text{ and the simplified result is } 1024 \)[/tex]
