Simplify ((4mn^-3)^2*-2m^3n^2)/(6m^-4n^6)

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

Step 1

Move $m^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2

Multiply $m^{3}$ by $m^{4}$ by adding the exponents.

Move $m^{4}$ .

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

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

$\frac{{(4 m n^{- 3})}^{2} \cdot (- 2 m^{4 + 3} n^{2})}{6 n^{6}}$

Add $4$ and $3$ .

$\frac{{(4 m n^{- 3})}^{2} \cdot (- 2 m^{7} n^{2})}{6 n^{6}}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 2$ and $6$ .

Factor $2$ out of ${(4 m n^{- 3})}^{2} \cdot (- 2 m^{7} n^{2})$ .

$\frac{2 ({(4 m n^{- 3})}^{2} \cdot (- m^{7} n^{2}))}{6 n^{6}}$

Cancel the common factors.

Factor $2$ out of $6 n^{6}$ .

$\frac{2 ({(4 m n^{- 3})}^{2} \cdot (- m^{7} n^{2}))}{2 (3 n^{6})}$

Cancel the common factor.

$\frac{2 ({(4 m n^{- 3})}^{2} \cdot (- m^{7} n^{2}))}{2 (3 n^{6})}$

Rewrite the expression.

$\frac{{(4 m n^{- 3})}^{2} \cdot (- m^{7} n^{2})}{3 n^{6}}$

Cancel the common factor of $n^{2}$ and $n^{6}$ .

Factor $n^{2}$ out of ${(4 m n^{- 3})}^{2} \cdot (- m^{7} n^{2})$ .

$\frac{n^{2} ({(4 m n^{- 3})}^{2} \cdot (- m^{7}))}{3 n^{6}}$

Cancel the common factors.

Factor $n^{2}$ out of $3 n^{6}$ .

$\frac{n^{2} ({(4 m n^{- 3})}^{2} \cdot (- m^{7}))}{n^{2} (3 n^{4})}$

Cancel the common factor.

$\frac{n^{2} ({(4 m n^{- 3})}^{2} \cdot (- m^{7}))}{n^{2} (3 n^{4})}$

Rewrite the expression.

$\frac{{(4 m n^{- 3})}^{2} \cdot (- m^{7})}{3 n^{4}}$

Step 4

Simplify the numerator.

Apply the product rule to $4 m n^{- 3}$ .

$\frac{{(4 m)}^{2} {(n^{- 3})}^{2} \cdot - 1 m^{7}}{3 n^{4}}$

Factor out negative.

$\frac{- {(4 m)}^{2} {(n^{- 3})}^{2} m^{7}}{3 n^{4}}$

Apply the product rule to $4 m$ .

$\frac{- (4^{2} m^{2}) {(n^{- 3})}^{2} m^{7}}{3 n^{4}}$

Raise $4$ to the power of $2$ .

$\frac{- (16 m^{2}) {(n^{- 3})}^{2} m^{7}}{3 n^{4}}$

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

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

$\frac{- (16 m^{2}) n^{- 3 \cdot 2} m^{7}}{3 n^{4}}$

Multiply $- 3$ by $2$ .

$\frac{- (16 m^{2}) n^{- 6} m^{7}}{3 n^{4}}$

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

$\frac{- 1 \cdot 16 m^{2} \frac{1}{n^{6}} m^{7}}{3 n^{4}}$

Combine exponents.

Multiply $- 1$ by $16$ .

$\frac{- 16 m^{2} \frac{1}{n^{6}} m^{7}}{3 n^{4}}$

Combine $- 16$ and $\frac{1}{n^{6}}$ .

$\frac{m^{2} \frac{- 16}{n^{6}} m^{7}}{3 n^{4}}$

Combine $m^{2}$ and $\frac{- 16}{n^{6}}$ .

$\frac{\frac{m^{2} \cdot - 16}{n^{6}} m^{7}}{3 n^{4}}$

Combine $\frac{m^{2} \cdot - 16}{n^{6}}$ and $m^{7}$ .

$\frac{\frac{m^{2} \cdot - 16 m^{7}}{n^{6}}}{3 n^{4}}$

Multiply $m^{2}$ by $m^{7}$ by adding the exponents.

Move $m^{7}$ .

$\frac{\frac{m^{7} m^{2} \cdot - 16}{n^{6}}}{3 n^{4}}$

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

$\frac{\frac{m^{7 + 2} \cdot - 16}{n^{6}}}{3 n^{4}}$

Add $7$ and $2$ .

$\frac{\frac{m^{9} \cdot - 16}{n^{6}}}{3 n^{4}}$

Move $- 16$ to the left of $m^{9}$ .

$\frac{\frac{- 16 \cdot m^{9}}{n^{6}}}{3 n^{4}}$

Move the negative in front of the fraction.

$\frac{- \frac{16 m^{9}}{n^{6}}}{3 n^{4}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$- \frac{16 m^{9}}{n^{6}} \cdot \frac{1}{3 n^{4}}$

Step 6

Multiply $- \frac{16 m^{9}}{n^{6}} \cdot \frac{1}{3 n^{4}}$ .

Multiply $\frac{1}{3 n^{4}}$ by $\frac{16 m^{9}}{n^{6}}$ .

$- \frac{16 m^{9}}{3 n^{4} n^{6}}$

Multiply $n^{4}$ by $n^{6}$ by adding the exponents.

Move $n^{6}$ .

$- \frac{16 m^{9}}{3 (n^{6} n^{4})}$

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

$- \frac{16 m^{9}}{3 n^{6 + 4}}$

Add $6$ and $4$ .

$- \frac{16 m^{9}}{3 n^{10}}$

Simplify ((4mn^-3)^2*-2m^3n^2)/(6m^-4n^6)

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