Basic Math Examples

$\frac{9 a^{2} - 4 b^{2}}{\sqrt{3 a} - \sqrt{2 b}}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Rewrite $9 a^{2}$ as ${(3 a)}^{2}$ .

$\frac{{(3 a)}^{2} - 4 b^{2}}{\sqrt{3 a} - \sqrt{2 b}}$

Step 1.2

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

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

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 a$ and $b = 2 b$ .

$\frac{(3 a + 2 b) (3 a - (2 b))}{\sqrt{3 a} - \sqrt{2 b}}$

Step 1.4

Multiply $2$ by $- 1$ .

$\frac{(3 a + 2 b) (3 a - 2 b)}{\sqrt{3 a} - \sqrt{2 b}}$

Step 2

Multiply $\frac{(3 a + 2 b) (3 a - 2 b)}{\sqrt{3 a} - \sqrt{2 b}}$ by $\frac{\sqrt{3 a} + \sqrt{2 b}}{\sqrt{3 a} + \sqrt{2 b}}$ .

$\frac{(3 a + 2 b) (3 a - 2 b)}{\sqrt{3 a} - \sqrt{2 b}} \cdot \frac{\sqrt{3 a} + \sqrt{2 b}}{\sqrt{3 a} + \sqrt{2 b}}$

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Multiply $\frac{(3 a + 2 b) (3 a - 2 b)}{\sqrt{3 a} - \sqrt{2 b}}$ by $\frac{\sqrt{3 a} + \sqrt{2 b}}{\sqrt{3 a} + \sqrt{2 b}}$ .

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{(\sqrt{3 a} - \sqrt{2 b}) (\sqrt{3 a} + \sqrt{2 b})}$

Step 3.2

Expand the denominator using the FOIL method.

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{{\sqrt{3 a}}^{2} + \sqrt{6 a b} - \sqrt{6 a b} - {\sqrt{2 b}}^{2}}$

Step 3.3

Simplify.

Tap for more steps...

Step 3.3.1

Rewrite ${\sqrt{3 a}}^{2}$ as $3 a$ .

Tap for more steps...

Step 3.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 a}$ as ${(3 a)}^{\frac{1}{2}}$ .

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{{({(3 a)}^{\frac{1}{2}})}^{2} - {\sqrt{2 b}}^{2}}$

Step 3.3.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{{(3 a)}^{\frac{1}{2} \cdot 2} - {\sqrt{2 b}}^{2}}$

Step 3.3.1.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{{(3 a)}^{\frac{2}{2}} - {\sqrt{2 b}}^{2}}$

Step 3.3.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1.4.1

Cancel the common factor.

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{{(3 a)}^{\frac{2}{2}} - {\sqrt{2 b}}^{2}}$

Step 3.3.1.4.2

Rewrite the expression.

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{{(3 a)}^{1} - {\sqrt{2 b}}^{2}}$

Step 3.3.1.5

Simplify.

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{3 a - {\sqrt{2 b}}^{2}}$

Step 3.3.2

Rewrite ${\sqrt{2 b}}^{2}$ as $2 b$ .

Tap for more steps...

Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 b}$ as ${(2 b)}^{\frac{1}{2}}$ .

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{3 a - {({(2 b)}^{\frac{1}{2}})}^{2}}$

Step 3.3.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{3 a - {(2 b)}^{\frac{1}{2} \cdot 2}}$

Step 3.3.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{3 a - {(2 b)}^{\frac{2}{2}}}$

Step 3.3.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.4.1

Cancel the common factor.

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{3 a - {(2 b)}^{\frac{2}{2}}}$

Step 3.3.2.4.2

Rewrite the expression.

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{3 a - {(2 b)}^{1}}$

Step 3.3.2.5

Simplify.

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{3 a - (2 b)}$

Step 3.3.3

Multiply $2$ by $- 1$ .

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{3 a - 2 b}$

Step 3.4

Cancel the common factor of $3 a - 2 b$ .

Tap for more steps...

Step 3.4.1

Cancel the common factor.

$\frac{(3 a + 2 b) (3 a - 2 b) (\sqrt{3 a} + \sqrt{2 b})}{3 a - 2 b}$

Step 3.4.2

Divide $(3 a + 2 b) (\sqrt{3 a} + \sqrt{2 b})$ by $1$ .

$(3 a + 2 b) (\sqrt{3 a} + \sqrt{2 b})$

Step 4

Expand $(3 a + 2 b) (\sqrt{3 a} + \sqrt{2 b})$ using the FOIL Method.

Tap for more steps...

Step 4.1

Apply the distributive property.

$3 a (\sqrt{3 a} + \sqrt{2 b}) + 2 b (\sqrt{3 a} + \sqrt{2 b})$

Step 4.2

Apply the distributive property.

$3 a \sqrt{3 a} + 3 a \sqrt{2 b} + 2 b (\sqrt{3 a} + \sqrt{2 b})$

Step 4.3

Apply the distributive property.

$3 a \sqrt{3 a} + 3 a \sqrt{2 b} + 2 b \sqrt{3 a} + 2 b \sqrt{2 b}$