Question

write the expression in simplest form. assume all variables are positive. $sqrt{81a^{7}b^{12}c^{9}} = \square$

Explanation:

Step1: Split root into product terms

$\sqrt{81a^7b^{12}c^9} = \sqrt{81} \cdot \sqrt{a^7} \cdot \sqrt{b^{12}} \cdot \sqrt{c^9}$

Step2: Simplify numerical square root

$\sqrt{81} = 9$

Step3: Simplify $a$-term root

$\sqrt{a^7} = \sqrt{a^6 \cdot a} = a^3\sqrt{a}$

Step4: Simplify $b$-term root

$\sqrt{b^{12}} = b^6$

Step5: Simplify $c$-term root

$\sqrt{c^9} = \sqrt{c^8 \cdot c} = c^4\sqrt{c}$

Step6: Combine all simplified terms

$9 \cdot a^3\sqrt{a} \cdot b^6 \cdot c^4\sqrt{c} = 9a^3b^6c^4\sqrt{ac}$

Answer:

$9a^3b^6c^4\sqrt{ac}$