Pre-Algebra Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Evaluate the exponent.

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

Step 3

Apply the distributive property.

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

Step 4

Combine using the product rule for radicals.

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

Step 5

Combine using the product rule for radicals.

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

Step 6

Simplify each term.

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Step 6.1

Multiply $2$ by $3$ .

$\frac{\sqrt{6} + \sqrt{3 \cdot 3}}{3}$

Step 6.2

Multiply $3$ by $3$ .

$\frac{\sqrt{6} + \sqrt{9}}{3}$

Step 6.3

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{6} + \sqrt{3^{2}}}{3}$

Step 6.4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{6} + 3}{3}$

Step 7

Split the fraction $\frac{\sqrt{6} + 3}{3}$ into two fractions.

$\frac{\sqrt{6}}{3} + \frac{3}{3}$

Step 8

Divide $3$ by $3$ .

$\frac{\sqrt{6}}{3} + 1$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6}}{3} + 1$

Decimal Form:

$1.81649658 \dots$