Simplify (2r^-5s^9t^6)^-3

${(2 r^{- 5} s^{9} t^{6})}^{- 3}$

Step 1

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

${(2 \frac{1}{r^{5}} s^{9} t^{6})}^{- 3}$

Step 2

Combine $2$ and $\frac{1}{r^{5}}$ .

${(\frac{2}{r^{5}} s^{9} t^{6})}^{- 3}$

Step 3

Combine $\frac{2}{r^{5}}$ and $s^{9}$ .

${(\frac{2 s^{9}}{r^{5}} t^{6})}^{- 3}$

Step 4

Combine $\frac{2 s^{9}}{r^{5}}$ and $t^{6}$ .

${(\frac{2 s^{9} t^{6}}{r^{5}})}^{- 3}$

Step 5

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

${(\frac{r^{5}}{2 s^{9} t^{6}})}^{3}$

Step 6

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

Apply the product rule to $\frac{r^{5}}{2 s^{9} t^{6}}$ .

$\frac{{(r^{5})}^{3}}{{(2 s^{9} t^{6})}^{3}}$

Apply the product rule to $2 s^{9} t^{6}$ .

$\frac{{(r^{5})}^{3}}{{(2 s^{9})}^{3} {(t^{6})}^{3}}$

Apply the product rule to $2 s^{9}$ .

$\frac{{(r^{5})}^{3}}{2^{3} {(s^{9})}^{3} {(t^{6})}^{3}}$

Step 7

Multiply the exponents in ${(r^{5})}^{3}$ .

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

$\frac{r^{5 \cdot 3}}{2^{3} {(s^{9})}^{3} {(t^{6})}^{3}}$

Multiply $5$ by $3$ .

$\frac{r^{15}}{2^{3} {(s^{9})}^{3} {(t^{6})}^{3}}$

Step 8

Simplify the denominator.

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

$\frac{r^{15}}{8 {(s^{9})}^{3} {(t^{6})}^{3}}$

Multiply the exponents in ${(s^{9})}^{3}$ .

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

$\frac{r^{15}}{8 s^{9 \cdot 3} {(t^{6})}^{3}}$

Multiply $9$ by $3$ .

$\frac{r^{15}}{8 s^{27} {(t^{6})}^{3}}$

Multiply the exponents in ${(t^{6})}^{3}$ .

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

$\frac{r^{15}}{8 s^{27} t^{6 \cdot 3}}$

Multiply $6$ by $3$ .

$\frac{r^{15}}{8 s^{27} t^{18}}$