Evaluate (-18m^2n)^2(-1/6mn^2)

${(- 18 m^{2} n)}^{2} (- \frac{1}{6} m n^{2})$

Step 1

Rewrite using the commutative property of multiplication.

$- \frac{1}{6} {(- 18 m^{2} n)}^{2} m n^{2}$

Step 2

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

Apply the product rule to $- 18 m^{2} n$ .

$- \frac{1}{6} ({(- 18 m^{2})}^{2} n^{2}) m n^{2}$

Apply the product rule to $- 18 m^{2}$ .

$- \frac{1}{6} ({(- 18)}^{2} {(m^{2})}^{2} n^{2}) m n^{2}$

Step 3

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

Move $n^{2}$ .

$- \frac{1}{6} ({(- 18)}^{2} {(m^{2})}^{2} (n^{2} n^{2})) m$

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

$- \frac{1}{6} ({(- 18)}^{2} {(m^{2})}^{2} n^{2 + 2}) m$

Add $2$ and $2$ .

$- \frac{1}{6} ({(- 18)}^{2} {(m^{2})}^{2} n^{4}) m$

Step 4

Raise $- 18$ to the power of $2$ .

$- \frac{1}{6} (324 {(m^{2})}^{2} n^{4}) m$

Step 5

Cancel the common factor of $6$ .

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$\frac{- 1}{6} (324 {(m^{2})}^{2} n^{4}) m$

Factor $6$ out of $324 {(m^{2})}^{2} n^{4}$ .

$\frac{- 1}{6} (6 (54 {(m^{2})}^{2} n^{4})) m$

Cancel the common factor.

$\frac{- 1}{6} (6 (54 {(m^{2})}^{2} n^{4})) m$

Rewrite the expression.

$- (54 {(m^{2})}^{2} n^{4}) m$

Step 6

Simplify the expression.

Multiply $54$ by $- 1$ .

$- 54 ({(m^{2})}^{2} n^{4}) m$

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

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

$- 54 (m^{2 \cdot 2} n^{4}) m$

Multiply $2$ by $2$ .

$- 54 (m^{4} n^{4}) m$

Step 7

Multiply $m^{4}$ by $m$ by adding the exponents.

Move $m$ .

$- 54 (m \cdot m^{4} n^{4})$

Multiply $m$ by $m^{4}$ .

Raise $m$ to the power of $1$ .

$- 54 (m^{1} m^{4} n^{4})$

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

$- 54 (m^{1 + 4} n^{4})$

Add $1$ and $4$ .

$- 54 (m^{5} n^{4})$

$- 54 m^{5} n^{4}$

Evaluate (-18m^2n)^2(-1/6mn^2)

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