Simplify (((12m^6n^4)/(mn))((m^2n)/(18m^5n^2)))^2

${((\frac{12 m^{6} n^{4}}{m n}) (\frac{m^{2} n}{18 m^{5} n^{2}}))}^{2}$

Step 1

Combine.

${(\frac{12 m^{6} n^{4} (m^{2} n)}{m n (18 m^{5} n^{2})})}^{2}$

Step 2

Multiply $m^{6}$ by $m^{2}$ by adding the exponents.

Move $m^{2}$ .

${(\frac{12 (m^{2} m^{6}) n^{4} n}{m n (18 m^{5} n^{2})})}^{2}$

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

${(\frac{12 m^{2 + 6} n^{4} n}{m n (18 m^{5} n^{2})})}^{2}$

Add $2$ and $6$ .

${(\frac{12 m^{8} n^{4} n}{m n (18 m^{5} n^{2})})}^{2}$

Step 3

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

Move $n$ .

${(\frac{12 m^{8} (n \cdot n^{4})}{m n (18 m^{5} n^{2})})}^{2}$

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

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

${(\frac{12 m^{8} (n^{1} n^{4})}{m n (18 m^{5} n^{2})})}^{2}$

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

${(\frac{12 m^{8} n^{1 + 4}}{m n (18 m^{5} n^{2})})}^{2}$

Add $1$ and $4$ .

${(\frac{12 m^{8} n^{5}}{m n (18 m^{5} n^{2})})}^{2}$

Step 4

Multiply $m$ by $m^{5}$ by adding the exponents.

Move $m^{5}$ .

${(\frac{12 m^{8} n^{5}}{m^{5} m n (18 n^{2})})}^{2}$

Multiply $m^{5}$ by $m$ .

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

${(\frac{12 m^{8} n^{5}}{m^{5} m^{1} n (18 n^{2})})}^{2}$

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

${(\frac{12 m^{8} n^{5}}{m^{5 + 1} n (18 n^{2})})}^{2}$

Add $5$ and $1$ .

${(\frac{12 m^{8} n^{5}}{m^{6} n (18 n^{2})})}^{2}$

Step 5

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

Move $n^{2}$ .

${(\frac{12 m^{8} n^{5}}{m^{6} (n^{2} n) \cdot 18})}^{2}$

Multiply $n^{2}$ by $n$ .

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

${(\frac{12 m^{8} n^{5}}{m^{6} (n^{2} n^{1}) \cdot 18})}^{2}$

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

${(\frac{12 m^{8} n^{5}}{m^{6} n^{2 + 1} \cdot 18})}^{2}$

Add $2$ and $1$ .

${(\frac{12 m^{8} n^{5}}{m^{6} n^{3} \cdot 18})}^{2}$

Step 6

Cancel the common factor of $12$ and $18$ .

Factor $6$ out of $12 m^{8} n^{5}$ .

${(\frac{6 (2 m^{8} n^{5})}{m^{6} n^{3} \cdot 18})}^{2}$

Cancel the common factors.

Factor $6$ out of $m^{6} n^{3} \cdot 18$ .

${(\frac{6 (2 m^{8} n^{5})}{6 (m^{6} n^{3} \cdot 3)})}^{2}$

Cancel the common factor.

${(\frac{6 (2 m^{8} n^{5})}{6 (m^{6} n^{3} \cdot 3)})}^{2}$

Rewrite the expression.

${(\frac{2 m^{8} n^{5}}{m^{6} n^{3} \cdot 3})}^{2}$

Step 7

Cancel the common factor of $m^{8}$ and $m^{6}$ .

Factor $m^{6}$ out of $2 m^{8} n^{5}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 8

Cancel the common factor of $n^{5}$ and $n^{3}$ .

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 9

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

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

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

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

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

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

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

Step 10

Simplify the numerator.

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

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

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

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

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

Multiply $2$ by $2$ .

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

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

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

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

Multiply $2$ by $2$ .

$\frac{4 m^{4} n^{4}}{3^{2}}$

Step 11

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

$\frac{4 m^{4} n^{4}}{9}$

Simplify (((12m^6n^4)/(mn))((m^2n)/(18m^5n^2)))^2

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