Simplify ((3r^-4s^2)/(2r^3s))^-3((6r^2s^3)/(8rs^5))^2

${(\frac{3 r^{- 4} s^{2}}{2 r^{3} s})}^{- 3} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

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

${(\frac{3 s^{2}}{2 r^{3} s r^{4}})}^{- 3} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

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

Move $r^{4}$ .

${(\frac{3 s^{2}}{2 (r^{4} r^{3}) s})}^{- 3} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

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

${(\frac{3 s^{2}}{2 r^{4 + 3} s})}^{- 3} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

Add $4$ and $3$ .

${(\frac{3 s^{2}}{2 r^{7} s})}^{- 3} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

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

Factor $s$ out of $3 s^{2}$ .

${(\frac{s (3 s)}{2 r^{7} s})}^{- 3} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

Cancel the common factors.

Factor $s$ out of $2 r^{7} s$ .

${(\frac{s (3 s)}{s (2 r^{7})})}^{- 3} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

Cancel the common factor.

${(\frac{s (3 s)}{s (2 r^{7})})}^{- 3} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

Rewrite the expression.

${(\frac{3 s}{2 r^{7}})}^{- 3} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{2 r^{7}}{3 s})}^{3} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

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

Apply the product rule to $\frac{2 r^{7}}{3 s}$ .

$\frac{{(2 r^{7})}^{3}}{{(3 s)}^{3}} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

Apply the product rule to $2 r^{7}$ .

$\frac{2^{3} {(r^{7})}^{3}}{{(3 s)}^{3}} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

Apply the product rule to $3 s$ .

$\frac{2^{3} {(r^{7})}^{3}}{3^{3} s^{3}} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

Simplify the numerator.

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

$\frac{8 {(r^{7})}^{3}}{3^{3} s^{3}} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

Multiply the exponents in ${(r^{7})}^{3}$ .

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

$\frac{8 r^{7 \cdot 3}}{3^{3} s^{3}} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

Multiply $7$ by $3$ .

$\frac{8 r^{21}}{3^{3} s^{3}} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

Reduce the expression by cancelling the common factors.

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

$\frac{8 r^{21}}{27 s^{3}} {(\frac{6 r^{2} s^{3}}{8 r s^{5}})}^{2}$

Cancel the common factor of $6$ and $8$ .

Factor $2$ out of $6 r^{2} s^{3}$ .

$\frac{8 r^{21}}{27 s^{3}} {(\frac{2 (3 r^{2} s^{3})}{8 r s^{5}})}^{2}$

Cancel the common factors.

Factor $2$ out of $8 r s^{5}$ .

$\frac{8 r^{21}}{27 s^{3}} {(\frac{2 (3 r^{2} s^{3})}{2 (4 r s^{5})})}^{2}$

Cancel the common factor.

$\frac{8 r^{21}}{27 s^{3}} {(\frac{2 (3 r^{2} s^{3})}{2 (4 r s^{5})})}^{2}$

Rewrite the expression.

$\frac{8 r^{21}}{27 s^{3}} {(\frac{3 r^{2} s^{3}}{4 r s^{5}})}^{2}$

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

Factor $r$ out of $3 r^{2} s^{3}$ .

$\frac{8 r^{21}}{27 s^{3}} {(\frac{r (3 r s^{3})}{4 r s^{5}})}^{2}$

Cancel the common factors.

Factor $r$ out of $4 r s^{5}$ .

$\frac{8 r^{21}}{27 s^{3}} {(\frac{r (3 r s^{3})}{r (4 s^{5})})}^{2}$

Cancel the common factor.

$\frac{8 r^{21}}{27 s^{3}} {(\frac{r (3 r s^{3})}{r (4 s^{5})})}^{2}$

Rewrite the expression.

$\frac{8 r^{21}}{27 s^{3}} {(\frac{3 r s^{3}}{4 s^{5}})}^{2}$

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

Factor $s^{3}$ out of $3 r s^{3}$ .

$\frac{8 r^{21}}{27 s^{3}} {(\frac{s^{3} (3 r)}{4 s^{5}})}^{2}$

Cancel the common factors.

Factor $s^{3}$ out of $4 s^{5}$ .

$\frac{8 r^{21}}{27 s^{3}} {(\frac{s^{3} (3 r)}{s^{3} (4 s^{2})})}^{2}$

Cancel the common factor.

$\frac{8 r^{21}}{27 s^{3}} {(\frac{s^{3} (3 r)}{s^{3} (4 s^{2})})}^{2}$

Rewrite the expression.

$\frac{8 r^{21}}{27 s^{3}} {(\frac{3 r}{4 s^{2}})}^{2}$

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

Apply the product rule to $\frac{3 r}{4 s^{2}}$ .

$\frac{8 r^{21}}{27 s^{3}} \cdot \frac{{(3 r)}^{2}}{{(4 s^{2})}^{2}}$

Apply the product rule to $3 r$ .

$\frac{8 r^{21}}{27 s^{3}} \cdot \frac{3^{2} r^{2}}{{(4 s^{2})}^{2}}$

Apply the product rule to $4 s^{2}$ .

$\frac{8 r^{21}}{27 s^{3}} \cdot \frac{3^{2} r^{2}}{4^{2} {(s^{2})}^{2}}$

Combine.

$\frac{8 r^{21} (3^{2} r^{2})}{27 s^{3} (4^{2} {(s^{2})}^{2})}$

Multiply $r^{21}$ by $r^{2}$ by adding the exponents.

Move $r^{2}$ .

$\frac{8 (r^{2} r^{21}) \cdot 3^{2}}{27 s^{3} (4^{2} {(s^{2})}^{2})}$

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

$\frac{8 r^{2 + 21} \cdot 3^{2}}{27 s^{3} (4^{2} {(s^{2})}^{2})}$

Add $2$ and $21$ .

$\frac{8 r^{23} \cdot 3^{2}}{27 s^{3} (4^{2} {(s^{2})}^{2})}$

Simplify the numerator.

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

$\frac{8 r^{23} \cdot 9}{27 s^{3} (4^{2} {(s^{2})}^{2})}$

Multiply $9$ by $8$ .

$\frac{72 r^{23}}{27 s^{3} (4^{2} {(s^{2})}^{2})}$

Simplify the denominator.

Remove unnecessary parentheses.

$\frac{72 r^{23}}{27 s^{3} \cdot 4^{2} {(s^{2})}^{2}}$

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

$\frac{72 r^{23}}{27 s^{3} \cdot 16 {(s^{2})}^{2}}$

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

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

$\frac{72 r^{23}}{27 s^{3} \cdot 16 s^{2 \cdot 2}}$

Multiply $2$ by $2$ .

$\frac{72 r^{23}}{27 s^{3} \cdot 16 s^{4}}$

Combine exponents.

Multiply $16$ by $27$ .

$\frac{72 r^{23}}{432 s^{3} s^{4}}$

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

Move $s^{4}$ .

$\frac{72 r^{23}}{432 (s^{4} s^{3})}$

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

$\frac{72 r^{23}}{432 s^{4 + 3}}$

Add $4$ and $3$ .

$\frac{72 r^{23}}{432 s^{7}}$

Cancel the common factor of $72$ and $432$ .

Factor $72$ out of $72 r^{23}$ .

$\frac{72 (r^{23})}{432 s^{7}}$

Cancel the common factors.

Factor $72$ out of $432 s^{7}$ .

$\frac{72 (r^{23})}{72 (6 s^{7})}$

Cancel the common factor.

$\frac{72 r^{23}}{72 (6 s^{7})}$

Rewrite the expression.

$\frac{r^{23}}{6 s^{7}}$

Simplify ((3r^-4s^2)/(2r^3s))^-3((6r^2s^3)/(8rs^5))^2

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