Algebra Examples

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

Step 1

Multiply $\sqrt{2 + \sqrt{3}}$ by $1$ .

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

Step 2

Multiply to rationalize the numerator.

$\frac{\sqrt{2 + \sqrt{3}} \sqrt{2 + \sqrt{3}}}{\sqrt{2 + \sqrt{3}}}$

Step 3

Simplify.

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Step 3.1

Raise $\sqrt{2 + \sqrt{3}}$ to the power of $1$ .

$\frac{{\sqrt{2 + \sqrt{3}}}^{1} \sqrt{2 + \sqrt{3}}}{\sqrt{2 + \sqrt{3}}}$

Step 3.2

Raise $\sqrt{2 + \sqrt{3}}$ to the power of $1$ .

$\frac{{\sqrt{2 + \sqrt{3}}}^{1} {\sqrt{2 + \sqrt{3}}}^{1}}{\sqrt{2 + \sqrt{3}}}$

Step 3.3

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

$\frac{{\sqrt{2 + \sqrt{3}}}^{1 + 1}}{\sqrt{2 + \sqrt{3}}}$

Step 3.4

Add $1$ and $1$ .

$\frac{{\sqrt{2 + \sqrt{3}}}^{2}}{\sqrt{2 + \sqrt{3}}}$

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Cancel the common factor of $2$ .

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Step 3.5.4.1

Cancel the common factor.

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

Step 3.5.4.2

Rewrite the expression.

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

Step 3.5.5

Simplify.

$\frac{2 + \sqrt{3}}{\sqrt{2 + \sqrt{3}}}$

Step 4

Multiply to rationalize the numerator.

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

Step 5

Simplify.

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Step 5.1

Expand the numerator using the FOIL method.

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

Step 5.2

Simplify.

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

Step 6

Apply the distributive property.

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

Step 7

Move $2$ to the left of $\sqrt{2 + \sqrt{3}}$ .

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

Step 8

Combine using the product rule for radicals.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.93185165 \dots$