Simplify -6r^-5s^-2*(-2r^3s^-2)^3

$- 6 r^{- 5} s^{- 2} \cdot {(- 2 r^{3} s^{- 2})}^{3}$

Step 1

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

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

Step 2

Combine $- 6$ and $\frac{1}{r^{5}}$ .

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

Step 3

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Multiply $\frac{1}{s^{2}}$ by $\frac{6}{r^{5}}$ .

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

Step 6

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

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

Step 7

Multiply $- 2 r^{3} \frac{1}{s^{2}}$ .

Combine $- 2$ and $\frac{1}{s^{2}}$ .

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

Combine $r^{3}$ and $\frac{- 2}{s^{2}}$ .

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

Step 8

Move $- 2$ to the left of $r^{3}$ .

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

Step 9

Move the negative in front of the fraction.

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

Step 10

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

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

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

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

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

Apply the product rule to $2 r^{3}$ .

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

Step 11

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

Move ${(- 1)}^{3}$ .

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

Multiply ${(- 1)}^{3}$ by $- 1$ .

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

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

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

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

Add $3$ and $1$ .

${(- 1)}^{4} \frac{6}{s^{2} r^{5}} \cdot \frac{2^{3} {(r^{3})}^{3}}{{(s^{2})}^{3}}$

Step 12

Simplify the numerator.

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

${(- 1)}^{4} \frac{6}{s^{2} r^{5}} \cdot \frac{8 {(r^{3})}^{3}}{{(s^{2})}^{3}}$

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

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

${(- 1)}^{4} \frac{6}{s^{2} r^{5}} \cdot \frac{8 r^{3 \cdot 3}}{{(s^{2})}^{3}}$

Multiply $3$ by $3$ .

${(- 1)}^{4} \frac{6}{s^{2} r^{5}} \cdot \frac{8 r^{9}}{{(s^{2})}^{3}}$

Step 13

Combine fractions.

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

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

${(- 1)}^{4} \frac{6}{s^{2} r^{5}} \cdot \frac{8 r^{9}}{s^{2 \cdot 3}}$

Multiply $2$ by $3$ .

${(- 1)}^{4} \frac{6}{s^{2} r^{5}} \cdot \frac{8 r^{9}}{s^{6}}$

Simplify the expression.

Raise $- 1$ to the power of $4$ .

$1 \frac{6}{s^{2} r^{5}} \cdot \frac{8 r^{9}}{s^{6}}$

Multiply $\frac{6}{s^{2} r^{5}}$ by $1$ .

$\frac{6}{s^{2} r^{5}} \cdot \frac{8 r^{9}}{s^{6}}$

Combine.

$\frac{6 (8 r^{9})}{s^{2} r^{5} s^{6}}$

Step 14

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

Move $s^{6}$ .

$\frac{6 (8 r^{9})}{s^{6} s^{2} r^{5}}$

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

$\frac{6 (8 r^{9})}{s^{6 + 2} r^{5}}$

Add $6$ and $2$ .

$\frac{6 (8 r^{9})}{s^{8} r^{5}}$

Step 15

Cancel the common factor of $r^{9}$ and $r^{5}$ .

Factor $r^{5}$ out of $6 (8 r^{9})$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{6 (8 r^{4})}{s^{8}}$

Step 16

Multiply $6$ by $8$ .

$\frac{48 r^{4}}{s^{8}}$

Simplify -6r^-5s^-2*(-2r^3s^-2)^3

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