Question

simplify the expression \\(\frac{1}{(2x)^{-2}}\\)

Explanation:

Step1: Recall the negative exponent rule

The negative exponent rule states that $a^{-n}=\frac{1}{a^{n}}$ (where $a
eq0$ and $n$ is a real number). So we can rewrite the denominator $(2x)^{-2}$ using this rule.
$(2x)^{-2}=\frac{1}{(2x)^{2}}$

Step2: Substitute the rewritten denominator into the original expression

The original expression is $\frac{1}{(2x)^{-2}}$, substituting the denominator we get:
$\frac{1}{\frac{1}{(2x)^{2}}}$

Step3: Divide by a fraction is multiply by its reciprocal

Dividing by a fraction $\frac{1}{b}$ is the same as multiplying by $b$. So $\frac{1}{\frac{1}{(2x)^{2}}}=1\times(2x)^{2}$

Step4: Expand $(2x)^{2}$

Using the power of a product rule $(ab)^{n}=a^{n}b^{n}$, we have $(2x)^{2}=2^{2}\times x^{2}=4x^{2}$

Answer:

$4x^{2}$