Question

Simplify the given expression. Assume that no denominator equals zero.

$\frac{{\left(3x^{-1}\right)}^{-2}}{{\left(3x^2\right)}^{-2}}$

Short answer

The final expression is $x^6$.

Step-by-step solution

Step 1. Define an expression.

A mathematical expression is a combinations of numbers and variables ordered with mathematical operations.

Step 2. Define power of a power rule.

The power of a power rule states that if a base raised to a power is being raised to another power, the exponents are multiplied and the base remains the same.

${\left(x^m\right)}^n=x^{mn}$

Where,$x$ is any non-zero real number.

Step 3. Define quotient of power rule.

The quotient of power rule is given by

$\frac{a^m}{a^n}=a^{m-n}$

Where,$a$ is any non-zero real number.

Step 4. Simplify the expression 3x−1−23x2−2.

Use the property of power of a power rule.

$$\begin{gathered} \frac{{\left(3x^{-1}\right)}^{-2}}{{\left(3x^2\right)}^{-2}}=\frac{{\left(3\right)}^{-2}{\left(x^{-1}\right)}^{-2}}{{\left(3\right)}^{-2}{\left(x^2\right)}^{-2}} \\ =\frac{{\left(x^{-1}\right)}^{-2}}{{\left(x^{-1}\right)}^{-2}} \\ =\frac{x^{-1\left(-2\right)}}{x^{2\left(-2\right)}} \\ =\frac{x^2}{x^{-4}} \end{gathered}$$

Use the property of quotient of power rule.

$\begin{array}{c}\frac{x^2}{x^{-4}}=x^{2-\left(-4\right)}\\ =x^{2+4}\\ =x^6\end{array}$

Hence,$\frac{{\left(3x^{-1}\right)}^{-2}}{{\left(3x^2\right)}^{-2}}=x^6$

Therefore the final expression is $x^6$.