To express the given expression [tex]\(\frac{\sqrt{a} \times a}{a^{-2}}\)[/tex] in the form [tex]\(a^k\)[/tex] and find the value of [tex]\(k\)[/tex], we can follow a series of algebraic steps to simplify it.
Let's start with the original expression:
[tex]\[
\frac{\sqrt{a} \times a}{a^{-2}}
\][/tex]
First, recognize that [tex]\(\sqrt{a}\)[/tex] can be written as a power of [tex]\(a\)[/tex]. Specifically, [tex]\(\sqrt{a}\)[/tex] is the same as [tex]\(a^{1/2}\)[/tex]. So, we can rewrite the expression as:
[tex]\[
\frac{a^{1/2} \times a}{a^{-2}}
\][/tex]
Next, simplify the product in the numerator. Recall that when multiplying powers with the same base, you add the exponents:
[tex]\[
a^{1/2} \times a = a^{1/2 + 1} = a^{1.5} \quad \text{(since } 1/2 + 1 = 1.5\text{)}
\][/tex]
Now, substitute this back into the expression:
[tex]\[
\frac{a^{1.5}}{a^{-2}}
\][/tex]
To simplify the division of powers with the same base, subtract the exponents:
[tex]\[
a^{1.5} \div a^{-2} = a^{1.5 - (-2)} = a^{1.5 + 2} = a^{3.5}
\][/tex]
Thus, the simplified expression can be written as [tex]\(a^{3.5}\)[/tex].
Comparing this with the form [tex]\(a^k\)[/tex], we see that:
[tex]\[
k = 3.5
\][/tex]
Therefore, the value of [tex]\(k\)[/tex] is:
[tex]\[
\boxed{3.5}
\][/tex]
