Algebra Examples

$\frac{\sqrt{a b}}{\sqrt{a} + \sqrt{b}}$

Step 1

Multiply $\frac{\sqrt{a b}}{\sqrt{a} + \sqrt{b}}$ by $\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}}$ .

$\frac{\sqrt{a b}}{\sqrt{a} + \sqrt{b}} \cdot \frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}}$

Step 2

Multiply $\frac{\sqrt{a b}}{\sqrt{a} + \sqrt{b}}$ by $\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} - \sqrt{b}}$ .

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

Step 3

Expand the denominator using the FOIL method.

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

Step 4

Simplify.

$\frac{\sqrt{a b} (\sqrt{a} - \sqrt{b})}{a - b}$

Step 5

Apply the distributive property.

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

Step 6

Multiply $\sqrt{a b} \sqrt{a}$ .

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Step 6.1

Combine using the product rule for radicals.

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

Step 6.2

Raise $a$ to the power of $1$ .

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

Step 6.3

Raise $a$ to the power of $1$ .

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

Step 6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.5

Add $1$ and $1$ .

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

Step 7

Rewrite using the commutative property of multiplication.

$\frac{\sqrt{a^{2} b} - \sqrt{a b} \sqrt{b}}{a - b}$

Step 8

Simplify each term.

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Step 8.1

Pull terms out from under the radical.

$\frac{a \sqrt{b} - \sqrt{a b} \sqrt{b}}{a - b}$

Step 8.2

Multiply $- \sqrt{a b} \sqrt{b}$ .

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Step 8.2.1

Combine using the product rule for radicals.

$\frac{a \sqrt{b} - \sqrt{b (a b)}}{a - b}$

Step 8.2.2

Raise $b$ to the power of $1$ .

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

Step 8.2.3

Raise $b$ to the power of $1$ .

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

Step 8.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{a \sqrt{b} - \sqrt{a b^{1 + 1}}}{a - b}$

Step 8.2.5

Add $1$ and $1$ .

$\frac{a \sqrt{b} - \sqrt{a b^{2}}}{a - b}$

Step 8.3

Reorder $a$ and $b^{2}$ .

$\frac{a \sqrt{b} - \sqrt{b^{2} a}}{a - b}$

Step 8.4

Pull terms out from under the radical.

$\frac{a \sqrt{b} - b \sqrt{a}}{a - b}$

Step 9

Simplify the numerator.

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Step 9.1

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

$\frac{a b^{\frac{1}{2}} - b \sqrt{a}}{a - b}$

Step 9.2

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

$\frac{a b^{\frac{1}{2}} - b a^{\frac{1}{2}}}{a - b}$

Step 9.3

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $a b^{\frac{1}{2}} - b a^{\frac{1}{2}}$ .

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Step 9.3.1

Move $b$ .

$\frac{a b^{\frac{1}{2}} - 1 a^{\frac{1}{2}} b}{a - b}$

Step 9.3.2

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $a b^{\frac{1}{2}}$ .

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

Step 9.3.3

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $- 1 a^{\frac{1}{2}} b$ .

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

Step 9.3.4

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} b^{\frac{1}{2}} (a^{\frac{1}{2}}) + a^{\frac{1}{2}} b^{\frac{1}{2}} (- b^{\frac{1}{2}})$ .

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