Certainly! Let's solve the expression step-by-step.
We start with the given mathematical expression:
[tex]\[
(2a)^{-1} \times 3a^2
\][/tex]
1. Understanding the inverse: The term [tex]\((2a)^{-1}\)[/tex] represents the reciprocal of [tex]\(2a\)[/tex]. Therefore:
[tex]\[
(2a)^{-1} = \frac{1}{2a}
\][/tex]
2. Substitute the reciprocal: Replace [tex]\((2a)^{-1}\)[/tex] with [tex]\(\frac{1}{2a}\)[/tex] in the original expression:
[tex]\[
\frac{1}{2a} \times 3a^2
\][/tex]
3. Simplify the multiplication: To simplify the expression, we need to multiply [tex]\(\frac{1}{2a}\)[/tex] by [tex]\(3a^2\)[/tex]:
[tex]\[
\frac{1}{2a} \times 3a^2
\][/tex]
4. Combine the fractions and algebraic terms:
[tex]\[
\frac{3a^2}{2a}
\][/tex]
5. Simplify the fraction: When simplifying [tex]\(\frac{3a^2}{2a}\)[/tex], we can cancel out [tex]\(a\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{3a^2}{2a} = \frac{3a \cdot a}{2a} = \frac{3a}{2}
\][/tex]
Thus, the simplified form of the expression [tex]\( (2 a)^{-1} \times 3 a^2 \)[/tex] is:
[tex]\[
\boxed{\frac{3a}{2}}
\][/tex]
So the expression simplifies to [tex]\(\frac{3a}{2}\)[/tex] at each step.
