Basic Math Examples

$\frac{4 - \sqrt{5}}{5 + \sqrt{3}}$

Step 1

Multiply $\frac{4 - \sqrt{5}}{5 + \sqrt{3}}$ by $\frac{5 - \sqrt{3}}{5 - \sqrt{3}}$ .

$\frac{4 - \sqrt{5}}{5 + \sqrt{3}} \cdot \frac{5 - \sqrt{3}}{5 - \sqrt{3}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{4 - \sqrt{5}}{5 + \sqrt{3}}$ by $\frac{5 - \sqrt{3}}{5 - \sqrt{3}}$ .

$\frac{(4 - \sqrt{5}) (5 - \sqrt{3})}{(5 + \sqrt{3}) (5 - \sqrt{3})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(4 - \sqrt{5}) (5 - \sqrt{3})}{25 - 5 \sqrt{3} + \sqrt{3} \cdot 5 - {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

$\frac{(4 - \sqrt{5}) (5 - \sqrt{3})}{22}$

Step 3

Expand $(4 - \sqrt{5}) (5 - \sqrt{3})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{4 (5 - \sqrt{3}) - \sqrt{5} (5 - \sqrt{3})}{22}$

Step 3.2

Apply the distributive property.

$\frac{4 \cdot 5 + 4 (- \sqrt{3}) - \sqrt{5} (5 - \sqrt{3})}{22}$

Step 3.3

Apply the distributive property.

$\frac{4 \cdot 5 + 4 (- \sqrt{3}) - \sqrt{5} \cdot 5 - \sqrt{5} (- \sqrt{3})}{22}$

Step 4

Simplify each term.

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Step 4.1

Multiply $4$ by $5$ .

$\frac{20 + 4 (- \sqrt{3}) - \sqrt{5} \cdot 5 - \sqrt{5} (- \sqrt{3})}{22}$

Step 4.2

Multiply $- 1$ by $4$ .

$\frac{20 - 4 \sqrt{3} - \sqrt{5} \cdot 5 - \sqrt{5} (- \sqrt{3})}{22}$

Step 4.3

Multiply $5$ by $- 1$ .

$\frac{20 - 4 \sqrt{3} - 5 \sqrt{5} - \sqrt{5} (- \sqrt{3})}{22}$

Step 4.4

Multiply $- \sqrt{5} (- \sqrt{3})$ .

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Step 4.4.1

Multiply $- 1$ by $- 1$ .

$\frac{20 - 4 \sqrt{3} - 5 \sqrt{5} + 1 \sqrt{5} \sqrt{3}}{22}$

Step 4.4.2

Multiply $\sqrt{5}$ by $1$ .

$\frac{20 - 4 \sqrt{3} - 5 \sqrt{5} + \sqrt{5} \sqrt{3}}{22}$

Step 4.4.3

Combine using the product rule for radicals.

$\frac{20 - 4 \sqrt{3} - 5 \sqrt{5} + \sqrt{5 \cdot 3}}{22}$

Step 4.4.4

Multiply $5$ by $3$ .

$\frac{20 - 4 \sqrt{3} - 5 \sqrt{5} + \sqrt{15}}{22}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{20 - 4 \sqrt{3} - 5 \sqrt{5} + \sqrt{15}}{22}$

Decimal Form:

$0.26202001 \dots$