Simplify 2ab( cube root of 192ab^2)-5 cube root of 81a^4b^5

$2 a b (\sqrt[3]{192 a b^{2}}) - 5 \sqrt[3]{81 a^{4} b^{5}}$

Step 1

Simplify each term.

Rewrite $192 a b^{2}$ as $4^{3} \cdot (3 a b^{2})$ .

Factor $64$ out of $192$ .

$2 a b \sqrt[3]{64 (3) a b^{2}} - 5 \sqrt[3]{81 a^{4} b^{5}}$

Rewrite $64$ as $4^{3}$ .

$2 a b \sqrt[3]{4^{3} \cdot 3 a b^{2}} - 5 \sqrt[3]{81 a^{4} b^{5}}$

Add parentheses.

$2 a b \sqrt[3]{4^{3} \cdot 3 (a b^{2})} - 5 \sqrt[3]{81 a^{4} b^{5}}$

Add parentheses.

$2 a b \sqrt[3]{4^{3} \cdot (3 a b^{2})} - 5 \sqrt[3]{81 a^{4} b^{5}}$

Pull terms out from under the radical.

$2 a b (4 \sqrt[3]{3 a b^{2}}) - 5 \sqrt[3]{81 a^{4} b^{5}}$

Multiply $4$ by $2$ .

$8 a b \sqrt[3]{3 a b^{2}} - 5 \sqrt[3]{81 a^{4} b^{5}}$

Rewrite $81 a^{4} b^{5}$ as ${(3 a b)}^{3} \cdot (3 a b^{2})$ .

Factor $27$ out of $81$ .

$8 a b \sqrt[3]{3 a b^{2}} - 5 \sqrt[3]{27 (3) a^{4} b^{5}}$

Rewrite $27$ as $3^{3}$ .

$8 a b \sqrt[3]{3 a b^{2}} - 5 \sqrt[3]{3^{3} \cdot 3 a^{4} b^{5}}$

Factor out $a^{3}$ .

$8 a b \sqrt[3]{3 a b^{2}} - 5 \sqrt[3]{3^{3} \cdot 3 (a^{3} a) b^{5}}$

Factor out $b^{3}$ .

$8 a b \sqrt[3]{3 a b^{2}} - 5 \sqrt[3]{3^{3} \cdot 3 (a^{3} a) (b^{3} b^{2})}$

Move $a$ .

$8 a b \sqrt[3]{3 a b^{2}} - 5 \sqrt[3]{3^{3} \cdot 3 (a^{3}) b^{3} a b^{2}}$

Move $3$ .

$8 a b \sqrt[3]{3 a b^{2}} - 5 \sqrt[3]{(3^{3} (a^{3})) b^{3} \cdot 3 a b^{2}}$

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

$8 a b \sqrt[3]{3 a b^{2}} - 5 \sqrt[3]{{(3 a b)}^{3} \cdot 3 a b^{2}}$

Add parentheses.

$8 a b \sqrt[3]{3 a b^{2}} - 5 \sqrt[3]{{(3 a b)}^{3} \cdot 3 (a b^{2})}$

Add parentheses.

$8 a b \sqrt[3]{3 a b^{2}} - 5 \sqrt[3]{{(3 a b)}^{3} \cdot (3 a b^{2})}$

Pull terms out from under the radical.

$8 a b \sqrt[3]{3 a b^{2}} - 5 (3 a b \sqrt[3]{3 a b^{2}})$

Multiply $3$ by $- 5$ .

$8 a b \sqrt[3]{3 a b^{2}} - 15 a b \sqrt[3]{3 a b^{2}}$

Step 2

Subtract $15 a b \sqrt[3]{3 a b^{2}}$ from $8 a b \sqrt[3]{3 a b^{2}}$ .

$- 7 a b \sqrt[3]{3 a b^{2}}$