Simplify ((3x^-1)^-2)/((3x^2)^-2)

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

Step 1

Simplify the expression.

Move ${(3 x^{- 1})}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Move ${(3 x^{2})}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2

Simplify the numerator.

Apply the product rule to $3 x^{2}$ .

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

Raise $3$ to the power of $2$ .

$\frac{9 {(x^{2})}^{2}}{{(3 x^{- 1})}^{2}}$

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{9 x^{2 \cdot 2}}{{(3 x^{- 1})}^{2}}$

Multiply $2$ by $2$ .

$\frac{9 x^{4}}{{(3 x^{- 1})}^{2}}$

Step 3

Simplify the denominator.

Apply the product rule to $3 x^{- 1}$ .

$\frac{9 x^{4}}{3^{2} {(x^{- 1})}^{2}}$

Raise $3$ to the power of $2$ .

$\frac{9 x^{4}}{9 {(x^{- 1})}^{2}}$

Multiply the exponents in ${(x^{- 1})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{9 x^{4}}{9 x^{- 1 \cdot 2}}$

Multiply $- 1$ by $2$ .

$\frac{9 x^{4}}{9 x^{- 2}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{9 x^{4}}{9 \frac{1}{x^{2}}}$

Step 4

Combine $9$ and $\frac{1}{x^{2}}$ .

$\frac{9 x^{4}}{\frac{9}{x^{2}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$9 x^{4} \frac{x^{2}}{9}$

Step 6

Cancel the common factor of $9$ .

Factor $9$ out of $9 x^{4}$ .

$9 (x^{4}) \frac{x^{2}}{9}$

Cancel the common factor.

$9 x^{4} \frac{x^{2}}{9}$

Rewrite the expression.

$x^{4} x^{2}$

Step 7

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{4 + 2}$

Add $4$ and $2$ .

$x^{6}$

Simplify ((3x^-1)^-2)/((3x^2)^-2)

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