Basic Math Examples

$\frac{6 - 9 \sqrt{7}}{8 + 4 \sqrt{10}}$

Step 1

Multiply $\frac{6 - 9 \sqrt{7}}{8 + 4 \sqrt{10}}$ by $\frac{8 - 4 \sqrt{10}}{8 - 4 \sqrt{10}}$ .

$\frac{6 - 9 \sqrt{7}}{8 + 4 \sqrt{10}} \cdot \frac{8 - 4 \sqrt{10}}{8 - 4 \sqrt{10}}$

Step 2

Simplify terms.

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Step 2.1

Multiply $\frac{6 - 9 \sqrt{7}}{8 + 4 \sqrt{10}}$ by $\frac{8 - 4 \sqrt{10}}{8 - 4 \sqrt{10}}$ .

$\frac{(6 - 9 \sqrt{7}) (8 - 4 \sqrt{10})}{(8 + 4 \sqrt{10}) (8 - 4 \sqrt{10})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(6 - 9 \sqrt{7}) (8 - 4 \sqrt{10})}{64 - 32 \sqrt{10} + 32 \sqrt{10} - 16 {\sqrt{10}}^{2}}$

Step 2.3

Simplify.

$\frac{(6 - 9 \sqrt{7}) (8 - 4 \sqrt{10})}{- 96}$

Step 2.4

Cancel the common factor of $6 - 9 \sqrt{7}$ and $- 96$ .

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Step 2.4.1

Factor $3$ out of $(6 - 9 \sqrt{7}) (8 - 4 \sqrt{10})$ .

$\frac{3 ((2 - 3 \sqrt{7}) (8 - 4 \sqrt{10}))}{- 96}$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $3$ out of $- 96$ .

$\frac{3 ((2 - 3 \sqrt{7}) (8 - 4 \sqrt{10}))}{3 (- 32)}$

Step 2.4.2.2

Cancel the common factor.

$\frac{3 ((2 - 3 \sqrt{7}) (8 - 4 \sqrt{10}))}{3 \cdot - 32}$

Step 2.4.2.3

Rewrite the expression.

$\frac{(2 - 3 \sqrt{7}) (8 - 4 \sqrt{10})}{- 32}$

Step 2.5

Cancel the common factor of $8 - 4 \sqrt{10}$ and $- 32$ .

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Step 2.5.1

Factor $4$ out of $(2 - 3 \sqrt{7}) (8 - 4 \sqrt{10})$ .

$\frac{4 ((2 - 3 \sqrt{7}) (2 - \sqrt{10}))}{- 32}$

Step 2.5.2

Cancel the common factors.

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Step 2.5.2.1

Factor $4$ out of $- 32$ .

$\frac{4 ((2 - 3 \sqrt{7}) (2 - \sqrt{10}))}{4 (- 8)}$

Step 2.5.2.2

Cancel the common factor.

$\frac{4 ((2 - 3 \sqrt{7}) (2 - \sqrt{10}))}{4 \cdot - 8}$

Step 2.5.2.3

Rewrite the expression.

$\frac{(2 - 3 \sqrt{7}) (2 - \sqrt{10})}{- 8}$

Step 3

Expand $(2 - 3 \sqrt{7}) (2 - \sqrt{10})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{2 (2 - \sqrt{10}) - 3 \sqrt{7} (2 - \sqrt{10})}{- 8}$

Step 3.2

Apply the distributive property.

$\frac{2 \cdot 2 + 2 (- \sqrt{10}) - 3 \sqrt{7} (2 - \sqrt{10})}{- 8}$

Step 3.3

Apply the distributive property.

$\frac{2 \cdot 2 + 2 (- \sqrt{10}) - 3 \sqrt{7} \cdot 2 - 3 \sqrt{7} (- \sqrt{10})}{- 8}$

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 $2$ .

$\frac{4 + 2 (- \sqrt{10}) - 3 \sqrt{7} \cdot 2 - 3 \sqrt{7} (- \sqrt{10})}{- 8}$

Step 4.1.2

Multiply $- 1$ by $2$ .

$\frac{4 - 2 \sqrt{10} - 3 \sqrt{7} \cdot 2 - 3 \sqrt{7} (- \sqrt{10})}{- 8}$

Step 4.1.3

Multiply $2$ by $- 3$ .

$\frac{4 - 2 \sqrt{10} - 6 \sqrt{7} - 3 \sqrt{7} (- \sqrt{10})}{- 8}$

Step 4.1.4

Multiply $- 3 \sqrt{7} (- \sqrt{10})$ .

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Step 4.1.4.1

Multiply $- 1$ by $- 3$ .

$\frac{4 - 2 \sqrt{10} - 6 \sqrt{7} + 3 \sqrt{7} \sqrt{10}}{- 8}$

Step 4.1.4.2

Combine using the product rule for radicals.

$\frac{4 - 2 \sqrt{10} - 6 \sqrt{7} + 3 \sqrt{10 \cdot 7}}{- 8}$

Step 4.1.4.3

Multiply $10$ by $7$ .

$\frac{4 - 2 \sqrt{10} - 6 \sqrt{7} + 3 \sqrt{70}}{- 8}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{4 - 2 \sqrt{10} - 6 \sqrt{7} + 3 \sqrt{70}}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{4 - 2 \sqrt{10} - 6 \sqrt{7} + 3 \sqrt{70}}{8}$

Decimal Form:

$- 0.86259220 \dots$