Simplify the Radical Expression fourth root of 243a^6b^8c^9

$\sqrt[4]{243 a^{6} b^{8} c^{9}}$

Step 1

Rewrite $243 a^{6} b^{8} c^{9}$ as ${(3 a b^{2} c^{2})}^{4} \cdot (3 a^{2} c)$ .

Factor $81$ out of $243$ .

$\sqrt[4]{81 (3) a^{6} b^{8} c^{9}}$

Rewrite $81$ as $3^{4}$ .

$\sqrt[4]{3^{4} \cdot 3 a^{6} b^{8} c^{9}}$

Factor out $a^{4}$ .

$\sqrt[4]{3^{4} \cdot 3 (a^{4} a^{2}) b^{8} c^{9}}$

Rewrite $b^{8}$ as ${(b^{2})}^{4}$ .

$\sqrt[4]{3^{4} \cdot 3 (a^{4} a^{2}) {(b^{2})}^{4} c^{9}}$

Factor out $c^{8}$ .

$\sqrt[4]{3^{4} \cdot 3 (a^{4} a^{2}) {(b^{2})}^{4} (c^{8} c)}$

Rewrite $c^{8}$ as ${(c^{2})}^{4}$ .

$\sqrt[4]{3^{4} \cdot 3 (a^{4} a^{2}) {(b^{2})}^{4} ({(c^{2})}^{4} c)}$

Move $a^{2}$ .

$\sqrt[4]{3^{4} \cdot 3 (a^{4}) {(b^{2})}^{4} {(c^{2})}^{4} a^{2} c}$

Move $3$ .

$\sqrt[4]{(3^{4} (a^{4})) {(b^{2})}^{4} {(c^{2})}^{4} \cdot 3 a^{2} c}$

Rewrite $(3^{4} (a^{4})) {(b^{2})}^{4} {(c^{2})}^{4}$ as ${(3 a b^{2} c^{2})}^{4}$ .

$\sqrt[4]{{(3 a b^{2} c^{2})}^{4} \cdot 3 a^{2} c}$

Add parentheses.

$\sqrt[4]{{(3 a b^{2} c^{2})}^{4} \cdot 3 (a^{2} c)}$

Add parentheses.

$\sqrt[4]{{(3 a b^{2} c^{2})}^{4} \cdot (3 a^{2} c)}$

Step 2

Pull terms out from under the radical.

$| 3 a b^{2} c^{2} | \sqrt[4]{3 a^{2} c}$

Step 3

Remove non-negative terms from the absolute value.

$3 b^{2} c^{2} | a | \sqrt[4]{3 a^{2} c}$