Algebra Examples

$\sqrt[3]{27 a^{6} b^{5}} - \sqrt{27 a^{5} b^{6}}$

Step 1

Rewrite $27 a^{6} b^{5}$ as ${(3 a^{2} b)}^{3} b^{2}$ .

Tap for more steps...

Step 1.1

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

$\sqrt[3]{3^{3} a^{6} b^{5}} - \sqrt{27 a^{5} b^{6}}$

Step 1.2

Rewrite $a^{6}$ as ${(a^{2})}^{3}$ .

$\sqrt[3]{3^{3} {(a^{2})}^{3} b^{5}} - \sqrt{27 a^{5} b^{6}}$

Step 1.3

Factor out $b^{3}$ .

$\sqrt[3]{3^{3} {(a^{2})}^{3} (b^{3} b^{2})} - \sqrt{27 a^{5} b^{6}}$

Step 1.4

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

$\sqrt[3]{{(3 a^{2} b)}^{3} b^{2}} - \sqrt{27 a^{5} b^{6}}$

Step 2

Pull terms out from under the radical.

$3 a^{2} b \sqrt[3]{b^{2}} - \sqrt{27 a^{5} b^{6}}$

Step 3

Rewrite $27 a^{5} b^{6}$ as ${(3 a^{2} b^{3})}^{2} \cdot (3 a)$ .

Tap for more steps...

Step 3.1

Factor $9$ out of $27$ .

$3 a^{2} b \sqrt[3]{b^{2}} - \sqrt{9 (3) a^{5} b^{6}}$

Step 3.2

Rewrite $9$ as $3^{2}$ .

$3 a^{2} b \sqrt[3]{b^{2}} - \sqrt{3^{2} \cdot 3 a^{5} b^{6}}$

Step 3.3

Factor out $a^{4}$ .

$3 a^{2} b \sqrt[3]{b^{2}} - \sqrt{3^{2} \cdot 3 (a^{4} a) b^{6}}$

Step 3.4

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

$3 a^{2} b \sqrt[3]{b^{2}} - \sqrt{3^{2} \cdot 3 ({(a^{2})}^{2} a) b^{6}}$

Step 3.5

Rewrite $b^{6}$ as ${(b^{3})}^{2}$ .

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

Step 3.6

Move $a$ .

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

Step 3.7

Move $3$ .

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

Step 3.8

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

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

Step 3.9

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

Multiply $3$ by $- 1$ .

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