Question

write the expression in simplest form. assume all variables are positive. $sqrt3{125r^{4}s^{9}t^{7}} = \square$

Explanation:

Step1: Split the cube root

$\sqrt[3]{125r^4s^9t^7} = \sqrt[3]{125} \cdot \sqrt[3]{r^4} \cdot \sqrt[3]{s^9} \cdot \sqrt[3]{t^7}$

Step2: Simplify constant cube root

$\sqrt[3]{125} = 5$, since $5^3=125$

Step3: Simplify $r$-term cube root

$\sqrt[3]{r^4} = \sqrt[3]{r^3 \cdot r} = r\sqrt[3]{r}$

Step4: Simplify $s$-term cube root

$\sqrt[3]{s^9} = s^3$, since $(s^3)^3=s^9$

Step5: Simplify $t$-term cube root

$\sqrt[3]{t^7} = \sqrt[3]{t^6 \cdot t} = t^2\sqrt[3]{t}$

Step6: Combine all simplified terms

$5 \cdot r\sqrt[3]{r} \cdot s^3 \cdot t^2\sqrt[3]{t} = 5r s^3 t^2 \sqrt[3]{r t}$

Answer:

$5rs^3t^2\sqrt[3]{rt}$