Question

simplify. assume all variables are positive.
\sqrt{5r^{6}s^{9}} \cdot \sqrt{10r^{2}s^{5}}
\sqrt{5r^{6}s^{9}} \cdot \sqrt{10r^{2}s^{5}} = \square
(type an exact answer, using radicals as needed.)

Explanation:

Step1: Combine the square roots

$\sqrt{5r^6s^9} \cdot \sqrt{10r^2s^5} = \sqrt{(5r^6s^9)(10r^2s^5)}$

Step2: Multiply terms inside the root

$\sqrt{5 \cdot 10 \cdot r^{6+2} \cdot s^{9+5}} = \sqrt{50r^8s^{14}}$

Step3: Factor into perfect squares

$\sqrt{25 \cdot 2 \cdot r^8 \cdot s^{14}} = \sqrt{25} \cdot \sqrt{r^8} \cdot \sqrt{s^{14}} \cdot \sqrt{2}$

Step4: Simplify perfect square roots

$5 \cdot r^4 \cdot s^7 \cdot \sqrt{2}$

Answer:

$5r^4s^7\sqrt{2}$