Precalculus Examples

$\frac{\sqrt{10} + \sqrt{3}}{\sqrt{10} - \sqrt{3}}$

Step 1

Multiply $\frac{\sqrt{10} + \sqrt{3}}{\sqrt{10} - \sqrt{3}}$ by $\frac{\sqrt{10} + \sqrt{3}}{\sqrt{10} + \sqrt{3}}$ .

$\frac{\sqrt{10} + \sqrt{3}}{\sqrt{10} - \sqrt{3}} \cdot \frac{\sqrt{10} + \sqrt{3}}{\sqrt{10} + \sqrt{3}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{10} + \sqrt{3}}{\sqrt{10} - \sqrt{3}}$ by $\frac{\sqrt{10} + \sqrt{3}}{\sqrt{10} + \sqrt{3}}$ .

$\frac{(\sqrt{10} + \sqrt{3}) (\sqrt{10} + \sqrt{3})}{(\sqrt{10} - \sqrt{3}) (\sqrt{10} + \sqrt{3})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{10} + \sqrt{3}) (\sqrt{10} + \sqrt{3})}{{\sqrt{10}}^{2} + \sqrt{30} - \sqrt{30} - {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{10} + \sqrt{3}) (\sqrt{10} + \sqrt{3})}{7}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $\sqrt{10} + \sqrt{3}$ to the power of $1$ .

$\frac{{(\sqrt{10} + \sqrt{3})}^{1} (\sqrt{10} + \sqrt{3})}{7}$

Step 3.2

Raise $\sqrt{10} + \sqrt{3}$ to the power of $1$ .

$\frac{{(\sqrt{10} + \sqrt{3})}^{1} {(\sqrt{10} + \sqrt{3})}^{1}}{7}$

Step 3.3

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

$\frac{{(\sqrt{10} + \sqrt{3})}^{1 + 1}}{7}$

Step 3.4

Add $1$ and $1$ .

$\frac{{(\sqrt{10} + \sqrt{3})}^{2}}{7}$

Step 4

Rewrite ${(\sqrt{10} + \sqrt{3})}^{2}$ as $(\sqrt{10} + \sqrt{3}) (\sqrt{10} + \sqrt{3})$ .

$\frac{(\sqrt{10} + \sqrt{3}) (\sqrt{10} + \sqrt{3})}{7}$

Step 5

Expand $(\sqrt{10} + \sqrt{3}) (\sqrt{10} + \sqrt{3})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{\sqrt{10} (\sqrt{10} + \sqrt{3}) + \sqrt{3} (\sqrt{10} + \sqrt{3})}{7}$

Step 5.2

Apply the distributive property.

$\frac{\sqrt{10} \sqrt{10} + \sqrt{10} \sqrt{3} + \sqrt{3} (\sqrt{10} + \sqrt{3})}{7}$

Step 5.3

Apply the distributive property.

$\frac{\sqrt{10} \sqrt{10} + \sqrt{10} \sqrt{3} + \sqrt{3} \sqrt{10} + \sqrt{3} \sqrt{3}}{7}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{10 \cdot 10} + \sqrt{10} \sqrt{3} + \sqrt{3} \sqrt{10} + \sqrt{3} \sqrt{3}}{7}$

Step 6.1.2

Multiply $10$ by $10$ .

$\frac{\sqrt{100} + \sqrt{10} \sqrt{3} + \sqrt{3} \sqrt{10} + \sqrt{3} \sqrt{3}}{7}$

Step 6.1.3

Rewrite $100$ as $10^{2}$ .

$\frac{\sqrt{10^{2}} + \sqrt{10} \sqrt{3} + \sqrt{3} \sqrt{10} + \sqrt{3} \sqrt{3}}{7}$

Step 6.1.4

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

$\frac{10 + \sqrt{10} \sqrt{3} + \sqrt{3} \sqrt{10} + \sqrt{3} \sqrt{3}}{7}$

Step 6.1.5

Combine using the product rule for radicals.

$\frac{10 + \sqrt{10 \cdot 3} + \sqrt{3} \sqrt{10} + \sqrt{3} \sqrt{3}}{7}$

Step 6.1.6

Multiply $10$ by $3$ .

$\frac{10 + \sqrt{30} + \sqrt{3} \sqrt{10} + \sqrt{3} \sqrt{3}}{7}$

Step 6.1.7

Combine using the product rule for radicals.

$\frac{10 + \sqrt{30} + \sqrt{3 \cdot 10} + \sqrt{3} \sqrt{3}}{7}$

Step 6.1.8

Multiply $3$ by $10$ .

$\frac{10 + \sqrt{30} + \sqrt{30} + \sqrt{3} \sqrt{3}}{7}$

Step 6.1.9

Combine using the product rule for radicals.

$\frac{10 + \sqrt{30} + \sqrt{30} + \sqrt{3 \cdot 3}}{7}$

Step 6.1.10

Multiply $3$ by $3$ .

$\frac{10 + \sqrt{30} + \sqrt{30} + \sqrt{9}}{7}$

Step 6.1.11

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

$\frac{10 + \sqrt{30} + \sqrt{30} + \sqrt{3^{2}}}{7}$

Step 6.1.12

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

$\frac{10 + \sqrt{30} + \sqrt{30} + 3}{7}$

Step 6.2

Add $10$ and $3$ .

$\frac{13 + \sqrt{30} + \sqrt{30}}{7}$

Step 6.3

Add $\sqrt{30}$ and $\sqrt{30}$ .

$\frac{13 + 2 \sqrt{30}}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{13 + 2 \sqrt{30}}{7}$

Decimal Form:

$3.42206445 \dots$