Finite Math Examples

$\frac{3 - 8 \sqrt{14}}{6 + 2 \sqrt{5}}$

Step 1

Multiply $\frac{3 - 8 \sqrt{14}}{6 + 2 \sqrt{5}}$ by $\frac{6 - 2 \sqrt{5}}{6 - 2 \sqrt{5}}$ .

$\frac{3 - 8 \sqrt{14}}{6 + 2 \sqrt{5}} \cdot \frac{6 - 2 \sqrt{5}}{6 - 2 \sqrt{5}}$

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{3 - 8 \sqrt{14}}{6 + 2 \sqrt{5}}$ by $\frac{6 - 2 \sqrt{5}}{6 - 2 \sqrt{5}}$ .

$\frac{(3 - 8 \sqrt{14}) (6 - 2 \sqrt{5})}{(6 + 2 \sqrt{5}) (6 - 2 \sqrt{5})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(3 - 8 \sqrt{14}) (6 - 2 \sqrt{5})}{36 - 12 \sqrt{5} + 12 \sqrt{5} - 4 {\sqrt{5}}^{2}}$

Step 2.3

Simplify.

$\frac{(3 - 8 \sqrt{14}) (6 - 2 \sqrt{5})}{16}$

Step 2.4

Cancel the common factor of $6 - 2 \sqrt{5}$ and $16$ .

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Step 2.4.1

Factor $2$ out of $(3 - 8 \sqrt{14}) (6 - 2 \sqrt{5})$ .

$\frac{2 ((3 - 8 \sqrt{14}) (3 - \sqrt{5}))}{16}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $2$ out of $16$ .

$\frac{2 ((3 - 8 \sqrt{14}) (3 - \sqrt{5}))}{2 (8)}$

Step 2.4.2.2

Cancel the common factor.

$\frac{2 ((3 - 8 \sqrt{14}) (3 - \sqrt{5}))}{2 \cdot 8}$

Step 2.4.2.3

Rewrite the expression.

$\frac{(3 - 8 \sqrt{14}) (3 - \sqrt{5})}{8}$

Step 3

Expand $(3 - 8 \sqrt{14}) (3 - \sqrt{5})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{3 (3 - \sqrt{5}) - 8 \sqrt{14} (3 - \sqrt{5})}{8}$

Step 3.2

Apply the distributive property.

$\frac{3 \cdot 3 + 3 (- \sqrt{5}) - 8 \sqrt{14} (3 - \sqrt{5})}{8}$

Step 3.3

Apply the distributive property.

$\frac{3 \cdot 3 + 3 (- \sqrt{5}) - 8 \sqrt{14} \cdot 3 - 8 \sqrt{14} (- \sqrt{5})}{8}$

Step 4

Simplify each term.

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Step 4.1

Multiply $3$ by $3$ .

$\frac{9 + 3 (- \sqrt{5}) - 8 \sqrt{14} \cdot 3 - 8 \sqrt{14} (- \sqrt{5})}{8}$

Step 4.2

Multiply $- 1$ by $3$ .

$\frac{9 - 3 \sqrt{5} - 8 \sqrt{14} \cdot 3 - 8 \sqrt{14} (- \sqrt{5})}{8}$

Step 4.3

Multiply $3$ by $- 8$ .

$\frac{9 - 3 \sqrt{5} - 24 \sqrt{14} - 8 \sqrt{14} (- \sqrt{5})}{8}$

Step 4.4

Multiply $- 8 \sqrt{14} (- \sqrt{5})$ .

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Step 4.4.1

Multiply $- 1$ by $- 8$ .

$\frac{9 - 3 \sqrt{5} - 24 \sqrt{14} + 8 \sqrt{14} \sqrt{5}}{8}$

Step 4.4.2

Combine using the product rule for radicals.

$\frac{9 - 3 \sqrt{5} - 24 \sqrt{14} + 8 \sqrt{5 \cdot 14}}{8}$

Step 4.4.3

Multiply $5$ by $14$ .

$\frac{9 - 3 \sqrt{5} - 24 \sqrt{14} + 8 \sqrt{70}}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{9 - 3 \sqrt{5} - 24 \sqrt{14} + 8 \sqrt{70}}{8}$

Decimal Form:

$- 2.57189738 \dots$