Sure, let's simplify the expression [tex]\((2a)^{-2}\)[/tex] step-by-step.
1. Understanding the Expression:
The expression [tex]\((2a)^{-2}\)[/tex] involves raising the term [tex]\(2a\)[/tex] to the power of [tex]\(-2\)[/tex].
2. Applying the Negative Exponent Rule:
The negative exponent rule states that [tex]\(x^{-n} = \frac{1}{x^n}\)[/tex]. Using this rule:
[tex]\[
(2a)^{-2} = \frac{1}{(2a)^2}
\][/tex]
3. Squaring the Term:
We need to square the term [tex]\(2a\)[/tex]. This involves squaring both the coefficient and the variable:
[tex]\[
(2a)^2 = 2^2 \cdot a^2
\][/tex]
Now, calculate [tex]\(2^2\)[/tex]:
[tex]\[
2^2 = 4
\][/tex]
Therefore:
[tex]\[
(2a)^2 = 4a^2
\][/tex]
4. Substituting Back:
Now, substitute the squared term back into the expression:
[tex]\[
\frac{1}{(2a)^2} = \frac{1}{4a^2}
\][/tex]
So, the simplified form of [tex]\((2a)^{-2}\)[/tex] is:
[tex]\[
\boxed{\frac{1}{4a^2}}
\][/tex]
