Simplify ((2m^-1n^4*3m^6n^2)/(4m^7n^6))^4

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

Step 1

Multiply $m^{- 1}$ by $m^{6}$ by adding the exponents.

Move $m^{6}$ .

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

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

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

Subtract $1$ from $6$ .

${(\frac{2 m^{5} n^{4} \cdot (3 n^{2})}{4 m^{7} n^{6}})}^{4}$

Step 2

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

Move $n^{2}$ .

${(\frac{2 m^{5} (n^{2} n^{4}) \cdot 3}{4 m^{7} n^{6}})}^{4}$

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

${(\frac{2 m^{5} n^{2 + 4} \cdot 3}{4 m^{7} n^{6}})}^{4}$

Add $2$ and $4$ .

${(\frac{2 m^{5} n^{6} \cdot 3}{4 m^{7} n^{6}})}^{4}$

Step 3

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 m^{5} n^{6} \cdot 3$ .

${(\frac{2 (m^{5} n^{6} \cdot 3)}{4 m^{7} n^{6}})}^{4}$

Cancel the common factors.

Factor $2$ out of $4 m^{7} n^{6}$ .

${(\frac{2 (m^{5} n^{6} \cdot 3)}{2 (2 m^{7} n^{6})})}^{4}$

Cancel the common factor.

${(\frac{2 (m^{5} n^{6} \cdot 3)}{2 (2 m^{7} n^{6})})}^{4}$

Rewrite the expression.

${(\frac{m^{5} n^{6} \cdot 3}{2 m^{7} n^{6}})}^{4}$

Step 4

Cancel the common factor of $m^{5}$ and $m^{7}$ .

Factor $m^{5}$ out of $m^{5} n^{6} \cdot 3$ .

${(\frac{m^{5} (n^{6} \cdot 3)}{2 m^{7} n^{6}})}^{4}$

Cancel the common factors.

Factor $m^{5}$ out of $2 m^{7} n^{6}$ .

${(\frac{m^{5} (n^{6} \cdot 3)}{m^{5} (2 m^{2} n^{6})})}^{4}$

Cancel the common factor.

${(\frac{m^{5} (n^{6} \cdot 3)}{m^{5} (2 m^{2} n^{6})})}^{4}$

Rewrite the expression.

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

Step 5

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

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

Apply the product rule to $\frac{3}{2 m^{2}}$ .

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

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

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

Step 7

Raise $3$ to the power of $4$ .

$\frac{81}{2^{4} {(m^{2})}^{4}}$

Step 8

Simplify the denominator.

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

$\frac{81}{16 {(m^{2})}^{4}}$

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

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

$\frac{81}{16 m^{2 \cdot 4}}$

Multiply $2$ by $4$ .

$\frac{81}{16 m^{8}}$

Simplify ((2m^-1n^4*3m^6n^2)/(4m^7n^6))^4

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