Simplify ((4k^3m^2)^3)/((5k^2m^-3)^-2)

$\frac{{(4 k^{3} m^{2})}^{3}}{{(5 k^{2} m^{- 3})}^{- 2}}$

Move ${(5 k^{2} m^{- 3})}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(4 k^{3} m^{2})}^{3} {(5 k^{2} m^{- 3})}^{2}$

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

Apply the product rule to $4 k^{3} m^{2}$ .

${(4 k^{3})}^{3} {(m^{2})}^{3} {(5 k^{2} m^{- 3})}^{2}$

Apply the product rule to $4 k^{3}$ .

$4^{3} {(k^{3})}^{3} {(m^{2})}^{3} {(5 k^{2} m^{- 3})}^{2}$

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

$64 {(k^{3})}^{3} {(m^{2})}^{3} {(5 k^{2} m^{- 3})}^{2}$

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

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

$64 k^{3 \cdot 3} {(m^{2})}^{3} {(5 k^{2} m^{- 3})}^{2}$

Multiply $3$ by $3$ .

$64 k^{9} {(m^{2})}^{3} {(5 k^{2} m^{- 3})}^{2}$

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

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

$64 k^{9} m^{2 \cdot 3} {(5 k^{2} m^{- 3})}^{2}$

Multiply $2$ by $3$ .

$64 k^{9} m^{6} {(5 k^{2} m^{- 3})}^{2}$

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

$64 k^{9} m^{6} {(5 k^{2} \frac{1}{m^{3}})}^{2}$

Multiply $5 k^{2} \frac{1}{m^{3}}$ .

Combine $5$ and $\frac{1}{m^{3}}$ .

$64 k^{9} m^{6} {(k^{2} \frac{5}{m^{3}})}^{2}$

Combine $k^{2}$ and $\frac{5}{m^{3}}$ .

$64 k^{9} m^{6} {(\frac{k^{2} \cdot 5}{m^{3}})}^{2}$

Move $5$ to the left of $k^{2}$ .

$64 k^{9} m^{6} {(\frac{5 \cdot k^{2}}{m^{3}})}^{2}$

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

Apply the product rule to $\frac{5 k^{2}}{m^{3}}$ .

$64 k^{9} m^{6} \frac{{(5 k^{2})}^{2}}{{(m^{3})}^{2}}$

Apply the product rule to $5 k^{2}$ .

$64 k^{9} m^{6} \frac{5^{2} {(k^{2})}^{2}}{{(m^{3})}^{2}}$

Simplify the numerator.

Raise $5$ to the power of $2$ .

$64 k^{9} m^{6} \frac{25 {(k^{2})}^{2}}{{(m^{3})}^{2}}$

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

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

$64 k^{9} m^{6} \frac{25 k^{2 \cdot 2}}{{(m^{3})}^{2}}$

Multiply $2$ by $2$ .

$64 k^{9} m^{6} \frac{25 k^{4}}{{(m^{3})}^{2}}$

Reduce the expression by cancelling the common factors.

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

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

$64 k^{9} m^{6} \frac{25 k^{4}}{m^{3 \cdot 2}}$

Multiply $3$ by $2$ .

$64 k^{9} m^{6} \frac{25 k^{4}}{m^{6}}$

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

Factor $m^{6}$ out of $64 k^{9} m^{6}$ .

$m^{6} (64 k^{9}) \frac{25 k^{4}}{m^{6}}$

Cancel the common factor.

$m^{6} (64 k^{9}) \frac{25 k^{4}}{m^{6}}$

Rewrite the expression.

$64 k^{9} (25 k^{4})$

Multiply $25$ by $64$ .

$1600 k^{9} k^{4}$

Multiply $k^{9}$ by $k^{4}$ by adding the exponents.

Move $k^{4}$ .

$1600 (k^{4} k^{9})$

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

$1600 k^{4 + 9}$

Add $4$ and $9$ .

$1600 k^{13}$

Simplify ((4k^3m^2)^3)/((5k^2m^-3)^-2)

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