Simplify ((6k^-2m^3)/(k^2))^-2

${(\frac{6 k^{- 2} m^{3}}{k^{2}})}^{- 2}$

Step 1

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

${(\frac{6 m^{3}}{k^{2} k^{2}})}^{- 2}$

Step 2

Multiply $k^{2}$ by $k^{2}$ by adding the exponents.

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

${(\frac{6 m^{3}}{k^{2 + 2}})}^{- 2}$

Add $2$ and $2$ .

${(\frac{6 m^{3}}{k^{4}})}^{- 2}$

Step 3

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{k^{4}}{6 m^{3}})}^{2}$

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{k^{4}}{6 m^{3}}$ .

$\frac{{(k^{4})}^{2}}{{(6 m^{3})}^{2}}$

Apply the product rule to $6 m^{3}$ .

$\frac{{(k^{4})}^{2}}{6^{2} {(m^{3})}^{2}}$

Step 5

Multiply the exponents in ${(k^{4})}^{2}$ .

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

$\frac{k^{4 \cdot 2}}{6^{2} {(m^{3})}^{2}}$

Multiply $4$ by $2$ .

$\frac{k^{8}}{6^{2} {(m^{3})}^{2}}$

Step 6

Simplify the denominator.

Raise $6$ to the power of $2$ .

$\frac{k^{8}}{36 {(m^{3})}^{2}}$

Multiply the exponents in ${(m^{3})}^{2}$ .

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

$\frac{k^{8}}{36 m^{3 \cdot 2}}$

Multiply $3$ by $2$ .

$\frac{k^{8}}{36 m^{6}}$