Basic Math Examples

\frac{\sqrt{2} + 1}{\sqrt{3} + 1} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}

$\frac{\sqrt{2} + 1}{\sqrt{3} + 1} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

Step 1

Simplify each term.

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Step 1.1

Multiply

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

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

\frac{\sqrt{3} - 1}{\sqrt{3} - 1}

$\frac{\sqrt{3} - 1}{\sqrt{3} - 1}$ .

\frac{\sqrt{2} + 1}{\sqrt{3} + 1} \cdot \frac{\sqrt{3} - 1}{\sqrt{3} - 1} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}

$\frac{\sqrt{2} + 1}{\sqrt{3} + 1} \cdot \frac{\sqrt{3} - 1}{\sqrt{3} - 1} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

Step 1.2

Multiply

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

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

\frac{\sqrt{3} - 1}{\sqrt{3} - 1}

$\frac{\sqrt{3} - 1}{\sqrt{3} - 1}$ .

\frac{(\sqrt{2} + 1) (\sqrt{3} - 1)}{(\sqrt{3} + 1) (\sqrt{3} - 1)} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}

$\frac{(\sqrt{2} + 1) (\sqrt{3} - 1)}{(\sqrt{3} + 1) (\sqrt{3} - 1)} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

Step 1.3

Expand the denominator using the FOIL method.

\frac{(\sqrt{2} + 1) (\sqrt{3} - 1)}{{\sqrt{3}}^{2} + \sqrt{3} \cdot - 1 + \sqrt{3} - 1} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}

$\frac{(\sqrt{2} + 1) (\sqrt{3} - 1)}{{\sqrt{3}}^{2} + \sqrt{3} \cdot - 1 + \sqrt{3} - 1} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

Step 1.4

Simplify.

\frac{(\sqrt{2} + 1) (\sqrt{3} - 1)}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}

$\frac{(\sqrt{2} + 1) (\sqrt{3} - 1)}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

Step 1.5

Expand

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

$(\sqrt{2} + 1) (\sqrt{3} - 1)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

\frac{\sqrt{2} (\sqrt{3} - 1) + 1 (\sqrt{3} - 1)}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}

$\frac{\sqrt{2} (\sqrt{3} - 1) + 1 (\sqrt{3} - 1)}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

Step 1.5.2

Apply the distributive property.

\frac{\sqrt{2} \sqrt{3} + \sqrt{2} \cdot - 1 + 1 (\sqrt{3} - 1)}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}

$\frac{\sqrt{2} \sqrt{3} + \sqrt{2} \cdot - 1 + 1 (\sqrt{3} - 1)}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

Step 1.5.3

Apply the distributive property.

\frac{\sqrt{2} \sqrt{3} + \sqrt{2} \cdot - 1 + 1 \sqrt{3} + 1 \cdot - 1}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}

$\frac{\sqrt{2} \sqrt{3} + \sqrt{2} \cdot - 1 + 1 \sqrt{3} + 1 \cdot - 1}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

\frac{\sqrt{2} \sqrt{3} + \sqrt{2} \cdot - 1 + 1 \sqrt{3} + 1 \cdot - 1}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}

$\frac{\sqrt{2} \sqrt{3} + \sqrt{2} \cdot - 1 + 1 \sqrt{3} + 1 \cdot - 1}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

Step 1.6

Simplify each term.

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Step 1.6.1

Combine using the product rule for radicals.

\frac{\sqrt{2 \cdot 3} + \sqrt{2} \cdot - 1 + 1 \sqrt{3} + 1 \cdot - 1}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}

$\frac{\sqrt{2 \cdot 3} + \sqrt{2} \cdot - 1 + 1 \sqrt{3} + 1 \cdot - 1}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

Step 1.6.2

Multiply

$2$ by

$3$ .

$\frac{\sqrt{6} + \sqrt{2} \cdot - 1 + 1 \sqrt{3} + 1 \cdot - 1}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

Step 1.6.3

Move

$- 1$ to the left of

$\sqrt{2}$ .

$\frac{\sqrt{6} - 1 \cdot \sqrt{2} + 1 \sqrt{3} + 1 \cdot - 1}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

Step 1.6.4

Rewrite

$- 1 \sqrt{2}$ as

$- \sqrt{2}$ .

$\frac{\sqrt{6} - \sqrt{2} + 1 \sqrt{3} + 1 \cdot - 1}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

Step 1.6.5

Multiply

$\sqrt{3}$ by

$1$ .

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} + 1 \cdot - 1}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

Step 1.6.6

Multiply

$- 1$ by

$1$ .

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} - \frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$

Step 1.7

Multiply

$\frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$ by

$\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}$ .

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} - (\frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}} \cdot \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}})$

Step 1.8

Multiply

$\frac{\sqrt{2} - 1}{\sqrt{5} - \sqrt{3}}$ by

$\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}$ .

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} - \frac{(\sqrt{2} - 1) (\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3}) (\sqrt{5} + \sqrt{3})}$

Step 1.9

Expand the denominator using the FOIL method.

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} - \frac{(\sqrt{2} - 1) (\sqrt{5} + \sqrt{3})}{{\sqrt{5}}^{2} + \sqrt{15} - \sqrt{15} - {\sqrt{3}}^{2}}$

Step 1.10

Simplify.

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} - \frac{(\sqrt{2} - 1) (\sqrt{5} + \sqrt{3})}{2}$

Step 1.11

Expand

$(\sqrt{2} - 1) (\sqrt{5} + \sqrt{3})$ using the FOIL Method.

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Step 1.11.1

Apply the distributive property.

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} - \frac{\sqrt{2} (\sqrt{5} + \sqrt{3}) - 1 (\sqrt{5} + \sqrt{3})}{2}$

Step 1.11.2

Apply the distributive property.

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} - \frac{\sqrt{2} \sqrt{5} + \sqrt{2} \sqrt{3} - 1 (\sqrt{5} + \sqrt{3})}{2}$

Step 1.11.3

Apply the distributive property.

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} - \frac{\sqrt{2} \sqrt{5} + \sqrt{2} \sqrt{3} - 1 \sqrt{5} - 1 \sqrt{3}}{2}$

Step 1.12

Simplify each term.

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Step 1.12.1

Combine using the product rule for radicals.

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} - \frac{\sqrt{2 \cdot 5} + \sqrt{2} \sqrt{3} - 1 \sqrt{5} - 1 \sqrt{3}}{2}$

Step 1.12.2

Multiply

$2$ by

$5$ .

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} - \frac{\sqrt{10} + \sqrt{2} \sqrt{3} - 1 \sqrt{5} - 1 \sqrt{3}}{2}$

Step 1.12.3

Combine using the product rule for radicals.

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} - \frac{\sqrt{10} + \sqrt{2 \cdot 3} - 1 \sqrt{5} - 1 \sqrt{3}}{2}$

Step 1.12.4

Multiply

$2$ by

$3$ .

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} - \frac{\sqrt{10} + \sqrt{6} - 1 \sqrt{5} - 1 \sqrt{3}}{2}$

Step 1.12.5

Rewrite

$- 1 \sqrt{5}$ as

$- \sqrt{5}$ .

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} - \frac{\sqrt{10} + \sqrt{6} - \sqrt{5} - 1 \sqrt{3}}{2}$

Step 1.12.6

Rewrite

$- 1 \sqrt{3}$ as

$- \sqrt{3}$ .

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} - \frac{\sqrt{10} + \sqrt{6} - \sqrt{5} - \sqrt{3}}{2}$

Step 2

Combine the numerators over the common denominator.

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1 - (\sqrt{10} + \sqrt{6} - \sqrt{5} - \sqrt{3})}{2}$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1 - \sqrt{10} - \sqrt{6} - - \sqrt{5} - - \sqrt{3}}{2}$

Step 3.2

Simplify.

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Step 3.2.1

Multiply

$- - \sqrt{5}$ .

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Step 3.2.1.1

Multiply

$- 1$ by

$- 1$ .

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1 - \sqrt{10} - \sqrt{6} + 1 \sqrt{5} - - \sqrt{3}}{2}$

Step 3.2.1.2

Multiply

$\sqrt{5}$ by

$1$ .

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1 - \sqrt{10} - \sqrt{6} + \sqrt{5} - - \sqrt{3}}{2}$

Step 3.2.2

Multiply

$- - \sqrt{3}$ .

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Step 3.2.2.1

Multiply

$- 1$ by

$- 1$ .

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1 - \sqrt{10} - \sqrt{6} + \sqrt{5} + 1 \sqrt{3}}{2}$

Step 3.2.2.2

Multiply

$\sqrt{3}$ by

$1$ .

$\frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1 - \sqrt{10} - \sqrt{6} + \sqrt{5} + \sqrt{3}}{2}$

Step 4

Simplify terms.

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Step 4.1

Subtract

$\sqrt{6}$ from

$\sqrt{6}$ .

$\frac{0 - \sqrt{2} + \sqrt{3} - 1 - \sqrt{10} + \sqrt{5} + \sqrt{3}}{2}$

Step 4.2

Subtract

$\sqrt{2}$ from

$0$ .

$\frac{- \sqrt{2} + \sqrt{3} - 1 - \sqrt{10} + \sqrt{5} + \sqrt{3}}{2}$

Step 4.3

Add

$\sqrt{3}$ and

$\sqrt{3}$ .

$\frac{- \sqrt{2} - 1 - \sqrt{10} + \sqrt{5} + 2 \sqrt{3}}{2}$

Step 4.4

Factor

$- 1$ out of

$- \sqrt{2}$ .

$\frac{- (\sqrt{2}) - 1 - \sqrt{10} + \sqrt{5} + 2 \sqrt{3}}{2}$

Step 4.5

Rewrite

$- 1$ as

$- 1 (1)$ .

$\frac{- (\sqrt{2}) - 1 (1) - \sqrt{10} + \sqrt{5} + 2 \sqrt{3}}{2}$

Step 4.6

Factor

$- 1$ out of

$- (\sqrt{2}) - 1 (1)$ .

$\frac{- (\sqrt{2} + 1) - \sqrt{10} + \sqrt{5} + 2 \sqrt{3}}{2}$

Step 4.7

Factor

$- 1$ out of

$- \sqrt{10}$ .

$\frac{- (\sqrt{2} + 1) - (\sqrt{10}) + \sqrt{5} + 2 \sqrt{3}}{2}$

Step 4.8

Factor

$- 1$ out of

$- (\sqrt{2} + 1) - (\sqrt{10})$ .

$\frac{- (\sqrt{2} + 1 + \sqrt{10}) + \sqrt{5} + 2 \sqrt{3}}{2}$

Step 4.9

Factor

$- 1$ out of

$\sqrt{5}$ .

$\frac{- (\sqrt{2} + 1 + \sqrt{10}) - 1 (- \sqrt{5}) + 2 \sqrt{3}}{2}$

Step 4.10

Factor

$- 1$ out of

$- (\sqrt{2} + 1 + \sqrt{10}) - 1 (- \sqrt{5})$ .

$\frac{- (\sqrt{2} + 1 + \sqrt{10} - \sqrt{5}) + 2 \sqrt{3}}{2}$

Step 4.11

Factor

$- 1$ out of

$2 \sqrt{3}$ .

$\frac{- (\sqrt{2} + 1 + \sqrt{10} - \sqrt{5}) - (- 2 \sqrt{3})}{2}$

Step 4.12

Factor

$- 1$ out of

$- (\sqrt{2} + 1 + \sqrt{10} - \sqrt{5}) - (- 2 \sqrt{3})$ .

$\frac{- (\sqrt{2} + 1 + \sqrt{10} - \sqrt{5} - 2 \sqrt{3})}{2}$

Step 4.13

Simplify the expression.

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Step 4.13.1

Rewrite

$- (\sqrt{2} + 1 + \sqrt{10} - \sqrt{5} - 2 \sqrt{3})$ as

$- 1 (\sqrt{2} + 1 + \sqrt{10} - \sqrt{5} - 2 \sqrt{3})$ .

$\frac{- 1 (\sqrt{2} + 1 + \sqrt{10} - \sqrt{5} - 2 \sqrt{3})}{2}$

Step 4.13.2

Move the negative in front of the fraction.

$- \frac{\sqrt{2} + 1 + \sqrt{10} - \sqrt{5} - 2 \sqrt{3}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{2} + 1 + \sqrt{10} - \sqrt{5} - 2 \sqrt{3}}{2}$

Decimal Form:

$0.06183918 \dots$