Linear Algebra Examples

$\frac{\sqrt{5} - 2 \sqrt{6}}{\sqrt{3} - 4 \sqrt{5}}$

Step 1

Multiply $\frac{\sqrt{5} - 2 \sqrt{6}}{\sqrt{3} - 4 \sqrt{5}}$ by $\frac{\sqrt{3} + 4 \sqrt{5}}{\sqrt{3} + 4 \sqrt{5}}$ .

$\frac{\sqrt{5} - 2 \sqrt{6}}{\sqrt{3} - 4 \sqrt{5}} \cdot \frac{\sqrt{3} + 4 \sqrt{5}}{\sqrt{3} + 4 \sqrt{5}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{5} - 2 \sqrt{6}}{\sqrt{3} - 4 \sqrt{5}}$ by $\frac{\sqrt{3} + 4 \sqrt{5}}{\sqrt{3} + 4 \sqrt{5}}$ .

$\frac{(\sqrt{5} - 2 \sqrt{6}) (\sqrt{3} + 4 \sqrt{5})}{(\sqrt{3} - 4 \sqrt{5}) (\sqrt{3} + 4 \sqrt{5})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{5} - 2 \sqrt{6}) (\sqrt{3} + 4 \sqrt{5})}{{\sqrt{3}}^{2} + 4 \sqrt{15} - 4 \sqrt{15} - 16 {\sqrt{5}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{5} - 2 \sqrt{6}) (\sqrt{3} + 4 \sqrt{5})}{- 77}$

Step 3

Expand $(\sqrt{5} - 2 \sqrt{6}) (\sqrt{3} + 4 \sqrt{5})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{5} (\sqrt{3} + 4 \sqrt{5}) - 2 \sqrt{6} (\sqrt{3} + 4 \sqrt{5})}{- 77}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{5} \sqrt{3} + \sqrt{5} (4 \sqrt{5}) - 2 \sqrt{6} (\sqrt{3} + 4 \sqrt{5})}{- 77}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{5} \sqrt{3} + \sqrt{5} (4 \sqrt{5}) - 2 \sqrt{6} \sqrt{3} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{5 \cdot 3} + \sqrt{5} (4 \sqrt{5}) - 2 \sqrt{6} \sqrt{3} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.2

Multiply $5$ by $3$ .

$\frac{\sqrt{15} + \sqrt{5} (4 \sqrt{5}) - 2 \sqrt{6} \sqrt{3} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.3

Multiply $\sqrt{5} (4 \sqrt{5})$ .

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Step 4.1.3.1

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{15} + 4 ({\sqrt{5}}^{1} \sqrt{5}) - 2 \sqrt{6} \sqrt{3} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{15} + 4 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) - 2 \sqrt{6} \sqrt{3} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{15} + 4 {\sqrt{5}}^{1 + 1} - 2 \sqrt{6} \sqrt{3} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.3.4

Add $1$ and $1$ .

$\frac{\sqrt{15} + 4 {\sqrt{5}}^{2} - 2 \sqrt{6} \sqrt{3} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.4

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt{15} + 4 {(5^{\frac{1}{2}})}^{2} - 2 \sqrt{6} \sqrt{3} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{15} + 4 \cdot 5^{\frac{1}{2} \cdot 2} - 2 \sqrt{6} \sqrt{3} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{15} + 4 \cdot 5^{\frac{2}{2}} - 2 \sqrt{6} \sqrt{3} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.4.4

Cancel the common factor of $2$ .

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Step 4.1.4.4.1

Cancel the common factor.

$\frac{\sqrt{15} + 4 \cdot 5^{\frac{2}{2}} - 2 \sqrt{6} \sqrt{3} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.4.4.2

Rewrite the expression.

$\frac{\sqrt{15} + 4 \cdot 5^{1} - 2 \sqrt{6} \sqrt{3} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.4.5

Evaluate the exponent.

$\frac{\sqrt{15} + 4 \cdot 5 - 2 \sqrt{6} \sqrt{3} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.5

Multiply $4$ by $5$ .

$\frac{\sqrt{15} + 20 - 2 \sqrt{6} \sqrt{3} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.6

Multiply $- 2 \sqrt{6} \sqrt{3}$ .

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Step 4.1.6.1

Combine using the product rule for radicals.

$\frac{\sqrt{15} + 20 - 2 \sqrt{3 \cdot 6} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.6.2

Multiply $3$ by $6$ .

$\frac{\sqrt{15} + 20 - 2 \sqrt{18} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.7

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 4.1.7.1

Factor $9$ out of $18$ .

$\frac{\sqrt{15} + 20 - 2 \sqrt{9 (2)} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.7.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{15} + 20 - 2 \sqrt{3^{2} \cdot 2} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.8

Pull terms out from under the radical.

$\frac{\sqrt{15} + 20 - 2 (3 \sqrt{2}) - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.9

Multiply $3$ by $- 2$ .

$\frac{\sqrt{15} + 20 - 6 \sqrt{2} - 2 \sqrt{6} (4 \sqrt{5})}{- 77}$

Step 4.1.10

Multiply $- 2 \sqrt{6} (4 \sqrt{5})$ .

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Step 4.1.10.1

Multiply $4$ by $- 2$ .

$\frac{\sqrt{15} + 20 - 6 \sqrt{2} - 8 \sqrt{6} \sqrt{5}}{- 77}$

Step 4.1.10.2

Combine using the product rule for radicals.

$\frac{\sqrt{15} + 20 - 6 \sqrt{2} - 8 \sqrt{5 \cdot 6}}{- 77}$

Step 4.1.10.3

Multiply $5$ by $6$ .

$\frac{\sqrt{15} + 20 - 6 \sqrt{2} - 8 \sqrt{30}}{- 77}$

Step 4.2

Move the negative in front of the fraction.

$- \frac{\sqrt{15} + 20 - 6 \sqrt{2} - 8 \sqrt{30}}{77}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{15} + 20 - 6 \sqrt{2} - 8 \sqrt{30}}{77}$

Decimal Form:

$0.36922211 \dots$