Algebra Examples

$\sqrt{a b}$

Step 1

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

$\sqrt{a} \sqrt{b}$

Step 2

Multiply to rationalize the numerator.

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $1$ .

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Cancel the common factor of $2$ .

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Step 3.5.4.1

Cancel the common factor.

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

Step 3.5.4.2

Rewrite the expression.

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

Step 3.5.5

Simplify.

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

Step 4

Multiply to rationalize the numerator.

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 5.5

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

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

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

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

Step 5.5.4

Cancel the common factor of $2$ .

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Step 5.5.4.1

Cancel the common factor.

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

Step 5.5.4.2

Rewrite the expression.

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

Step 5.5.5

Simplify.

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

Step 5.6

Combine using the product rule for radicals.

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