Simplify (24m^8n^5)/((-m^2n)^3+5m^6n^3)*(-3mn^4)^2

$\frac{24 m^{8} n^{5}}{{(- m^{2} n)}^{3} + 5 m^{6} n^{3}} \cdot {(- 3 m n^{4})}^{2}$

Simplify the denominator.

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

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

$\frac{24 m^{8} n^{5}}{{(- m^{2})}^{3} n^{3} + 5 m^{6} n^{3}} \cdot {(- 3 m n^{4})}^{2}$

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

$\frac{24 m^{8} n^{5}}{{(- 1)}^{3} {(m^{2})}^{3} n^{3} + 5 m^{6} n^{3}} \cdot {(- 3 m n^{4})}^{2}$

Raise $- 1$ to the power of $3$ .

$\frac{24 m^{8} n^{5}}{- {(m^{2})}^{3} n^{3} + 5 m^{6} n^{3}} \cdot {(- 3 m n^{4})}^{2}$

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

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

$\frac{24 m^{8} n^{5}}{- m^{2 \cdot 3} n^{3} + 5 m^{6} n^{3}} \cdot {(- 3 m n^{4})}^{2}$

Multiply $2$ by $3$ .

$\frac{24 m^{8} n^{5}}{- m^{6} n^{3} + 5 m^{6} n^{3}} \cdot {(- 3 m n^{4})}^{2}$

Add $- m^{6} n^{3}$ and $5 m^{6} n^{3}$ .

$\frac{24 m^{8} n^{5}}{4 m^{6} n^{3}} \cdot {(- 3 m n^{4})}^{2}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $24$ and $4$ .

Factor $4$ out of $24 m^{8} n^{5}$ .

$\frac{4 (6 m^{8} n^{5})}{4 m^{6} n^{3}} \cdot {(- 3 m n^{4})}^{2}$

Cancel the common factors.

Factor $4$ out of $4 m^{6} n^{3}$ .

$\frac{4 (6 m^{8} n^{5})}{4 (m^{6} n^{3})} \cdot {(- 3 m n^{4})}^{2}$

Cancel the common factor.

$\frac{4 (6 m^{8} n^{5})}{4 (m^{6} n^{3})} \cdot {(- 3 m n^{4})}^{2}$

Rewrite the expression.

$\frac{6 m^{8} n^{5}}{m^{6} n^{3}} \cdot {(- 3 m n^{4})}^{2}$

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factors.

Multiply by $1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $6 m^{2} n^{2}$ by $1$ .

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

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

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

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

Apply the product rule to $- 3 m$ .

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

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

Move $m^{2}$ .

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

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

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

Add $2$ and $2$ .

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

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

$6 m^{4} n^{2} \cdot (9 {(n^{4})}^{2})$

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

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

$6 m^{4} n^{2} \cdot (9 n^{4 \cdot 2})$

Multiply $4$ by $2$ .

$6 m^{4} n^{2} \cdot (9 n^{8})$

Multiply $n^{2}$ by $n^{8}$ by adding the exponents.

Move $n^{8}$ .

$6 m^{4} (n^{8} n^{2}) \cdot 9$

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

$6 m^{4} n^{8 + 2} \cdot 9$

Add $8$ and $2$ .

$6 m^{4} n^{10} \cdot 9$

Multiply $9$ by $6$ .

$54 m^{4} n^{10}$

Simplify (24m^8n^5)/((-m^2n)^3+5m^6n^3)*(-3mn^4)^2

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