Simplify cube root of r^-6*r^-3

$\sqrt[3]{r^{- 6} \cdot r^{- 3}}$

Step 1

Multiply $r^{- 6}$ by $r^{- 3}$ by adding the exponents.

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

$\sqrt[3]{r^{- 6 - 3}}$

Subtract $3$ from $- 6$ .

$\sqrt[3]{r^{- 9}}$

Step 2

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

$\sqrt[3]{\frac{1}{r^{9}}}$

Step 3

Rewrite $1$ as $1^{3}$ .

$\sqrt[3]{\frac{1^{3}}{r^{9}}}$

Step 4

Rewrite $r^{9}$ as ${(r^{3})}^{3}$ .

$\sqrt[3]{\frac{1^{3}}{{(r^{3})}^{3}}}$

Step 5

Rewrite $\frac{1^{3}}{{(r^{3})}^{3}}$ as ${(\frac{1}{r^{3}})}^{3}$ .

$\sqrt[3]{{(\frac{1}{r^{3}})}^{3}}$

Step 6

Pull terms out from under the radical, assuming real numbers.

$\frac{1}{r^{3}}$