Question

simplify. assume all variables are positive.
\sqrt{2r^{8}s^{5}} \cdot \sqrt{10r^{2}s^{3}}
\sqrt{2r^{8}s^{5}} \cdot \sqrt{10r^{2}s^{3}} = \square
(type an exact answer, using radicals as needed.)

Explanation:

Step1: Combine the square roots

$\sqrt{2r^8s^5} \cdot \sqrt{10r^2s^3} = \sqrt{(2r^8s^5)(10r^2s^3)}$

Step2: Multiply the terms inside

$\sqrt{2 \cdot 10 \cdot r^{8+2} \cdot s^{5+3}} = \sqrt{20r^{10}s^8}$

Step3: Factor into perfect squares

$\sqrt{4 \cdot 5 \cdot (r^5)^2 \cdot (s^4)^2} = \sqrt{4} \cdot \sqrt{(r^5)^2} \cdot \sqrt{(s^4)^2} \cdot \sqrt{5}$

Step4: Simplify each perfect square

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

Answer:

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