Basic Math Examples

$\frac{2 - 4 \sqrt{2}}{4 + 5 \sqrt{3}}$

Step 1

Multiply $\frac{2 - 4 \sqrt{2}}{4 + 5 \sqrt{3}}$ by $\frac{4 - 5 \sqrt{3}}{4 - 5 \sqrt{3}}$ .

$\frac{2 - 4 \sqrt{2}}{4 + 5 \sqrt{3}} \cdot \frac{4 - 5 \sqrt{3}}{4 - 5 \sqrt{3}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{2 - 4 \sqrt{2}}{4 + 5 \sqrt{3}}$ by $\frac{4 - 5 \sqrt{3}}{4 - 5 \sqrt{3}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(2 - 4 \sqrt{2}) (4 - 5 \sqrt{3})}{16 - 20 \sqrt{3} + 20 \sqrt{3} - 25 {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

$\frac{(2 - 4 \sqrt{2}) (4 - 5 \sqrt{3})}{- 59}$

Step 3

Expand $(2 - 4 \sqrt{2}) (4 - 5 \sqrt{3})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{2 (4 - 5 \sqrt{3}) - 4 \sqrt{2} (4 - 5 \sqrt{3})}{- 59}$

Step 3.2

Apply the distributive property.

$\frac{2 \cdot 4 + 2 (- 5 \sqrt{3}) - 4 \sqrt{2} (4 - 5 \sqrt{3})}{- 59}$

Step 3.3

Apply the distributive property.

$\frac{2 \cdot 4 + 2 (- 5 \sqrt{3}) - 4 \sqrt{2} \cdot 4 - 4 \sqrt{2} (- 5 \sqrt{3})}{- 59}$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $2$ by $4$ .

$\frac{8 + 2 (- 5 \sqrt{3}) - 4 \sqrt{2} \cdot 4 - 4 \sqrt{2} (- 5 \sqrt{3})}{- 59}$

Step 4.1.2

Multiply $- 5$ by $2$ .

$\frac{8 - 10 \sqrt{3} - 4 \sqrt{2} \cdot 4 - 4 \sqrt{2} (- 5 \sqrt{3})}{- 59}$

Step 4.1.3

Multiply $4$ by $- 4$ .

$\frac{8 - 10 \sqrt{3} - 16 \sqrt{2} - 4 \sqrt{2} (- 5 \sqrt{3})}{- 59}$

Step 4.1.4

Multiply $- 4 \sqrt{2} (- 5 \sqrt{3})$ .

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Step 4.1.4.1

Multiply $- 5$ by $- 4$ .

$\frac{8 - 10 \sqrt{3} - 16 \sqrt{2} + 20 \sqrt{2} \sqrt{3}}{- 59}$

Step 4.1.4.2

Combine using the product rule for radicals.

$\frac{8 - 10 \sqrt{3} - 16 \sqrt{2} + 20 \sqrt{3 \cdot 2}}{- 59}$

Step 4.1.4.3

Multiply $3$ by $2$ .

$\frac{8 - 10 \sqrt{3} - 16 \sqrt{2} + 20 \sqrt{6}}{- 59}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{8 - 10 \sqrt{3} - 16 \sqrt{2} + 20 \sqrt{6}}{59}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{8 - 10 \sqrt{3} - 16 \sqrt{2} + 20 \sqrt{6}}{59}$

Decimal Form:

$- 0.28884525 \dots$