Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite $\sqrt{\frac{a}{b}}$ as $\frac{\sqrt{a}}{\sqrt{b}}$ .

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

Step 1.2

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

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

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

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

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

Step 1.3.2

Raise $\sqrt{b}$ to the power of $1$ .

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

Step 1.3.3

Raise $\sqrt{b}$ to the power of $1$ .

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

Step 1.3.4

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

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

Step 1.3.5

Add $1$ and $1$ .

$\frac{\sqrt{a} \sqrt{b}}{{\sqrt{b}}^{2}} + \frac{a}{b} \cdot \sqrt{\frac{b}{a}}$

Step 1.3.6

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

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Step 1.3.6.1

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

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

Step 1.3.6.2

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

$\frac{\sqrt{a} \sqrt{b}}{b^{\frac{1}{2} \cdot 2}} + \frac{a}{b} \cdot \sqrt{\frac{b}{a}}$

Step 1.3.6.3

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

$\frac{\sqrt{a} \sqrt{b}}{b^{\frac{2}{2}}} + \frac{a}{b} \cdot \sqrt{\frac{b}{a}}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{a} \sqrt{b}}{b^{\frac{2}{2}}} + \frac{a}{b} \cdot \sqrt{\frac{b}{a}}$

Step 1.3.6.4.2

Rewrite the expression.

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

Step 1.3.6.5

Simplify.

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

Step 1.4

Combine using the product rule for radicals.

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

Step 1.5

Rewrite $\sqrt{\frac{b}{a}}$ as $\frac{\sqrt{b}}{\sqrt{a}}$ .

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

Step 1.6

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

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

Step 1.7

Combine and simplify the denominator.

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Step 1.7.1

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

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

Step 1.7.2

Raise $\sqrt{a}$ to the power of $1$ .

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

Step 1.7.3

Raise $\sqrt{a}$ to the power of $1$ .

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

Step 1.7.4

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

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

Step 1.7.5

Add $1$ and $1$ .

$\frac{\sqrt{a b}}{b} + \frac{a}{b} \cdot \frac{\sqrt{b} \sqrt{a}}{{\sqrt{a}}^{2}}$

Step 1.7.6

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

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Step 1.7.6.1

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

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

Step 1.7.6.2

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

$\frac{\sqrt{a b}}{b} + \frac{a}{b} \cdot \frac{\sqrt{b} \sqrt{a}}{a^{\frac{1}{2} \cdot 2}}$

Step 1.7.6.3

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

$\frac{\sqrt{a b}}{b} + \frac{a}{b} \cdot \frac{\sqrt{b} \sqrt{a}}{a^{\frac{2}{2}}}$

Step 1.7.6.4

Cancel the common factor of $2$ .

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Step 1.7.6.4.1

Cancel the common factor.

$\frac{\sqrt{a b}}{b} + \frac{a}{b} \cdot \frac{\sqrt{b} \sqrt{a}}{a^{\frac{2}{2}}}$

Step 1.7.6.4.2

Rewrite the expression.

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

Step 1.7.6.5

Simplify.

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

Step 1.8

Combine using the product rule for radicals.

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

Step 1.9

Cancel the common factor of $a$ .

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Step 1.9.1

Cancel the common factor.

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

Step 1.9.2

Rewrite the expression.

$\frac{\sqrt{a b}}{b} + \frac{1}{b} \cdot \sqrt{b a}$

Step 1.10

Combine $\frac{1}{b}$ and $\sqrt{b a}$ .

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Add $\sqrt{a b}$ and $\sqrt{b a}$ .

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Step 3.1

Reorder $b$ and $a$ .

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

Step 3.2

Add $\sqrt{a b}$ and $\sqrt{a b}$ .

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