Linear Algebra Examples

$\frac{\sqrt{5} - \sqrt{7}}{\sqrt{35} - \sqrt{34}}$

Step 1

Multiply $\frac{\sqrt{5} - \sqrt{7}}{\sqrt{35} - \sqrt{34}}$ by $\frac{\sqrt{35} + \sqrt{34}}{\sqrt{35} + \sqrt{34}}$ .

$\frac{\sqrt{5} - \sqrt{7}}{\sqrt{35} - \sqrt{34}} \cdot \frac{\sqrt{35} + \sqrt{34}}{\sqrt{35} + \sqrt{34}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{5} - \sqrt{7}}{\sqrt{35} - \sqrt{34}}$ by $\frac{\sqrt{35} + \sqrt{34}}{\sqrt{35} + \sqrt{34}}$ .

$\frac{(\sqrt{5} - \sqrt{7}) (\sqrt{35} + \sqrt{34})}{(\sqrt{35} - \sqrt{34}) (\sqrt{35} + \sqrt{34})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{5} - \sqrt{7}) (\sqrt{35} + \sqrt{34})}{{\sqrt{35}}^{2} + \sqrt{1190} - \sqrt{1190} - {\sqrt{34}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{5} - \sqrt{7}) (\sqrt{35} + \sqrt{34})}{1}$

Step 2.4

Divide $(\sqrt{5} - \sqrt{7}) (\sqrt{35} + \sqrt{34})$ by $1$ .

$(\sqrt{5} - \sqrt{7}) (\sqrt{35} + \sqrt{34})$

Step 3

Expand $(\sqrt{5} - \sqrt{7}) (\sqrt{35} + \sqrt{34})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\sqrt{5} (\sqrt{35} + \sqrt{34}) - \sqrt{7} (\sqrt{35} + \sqrt{34})$

Step 3.2

Apply the distributive property.

$\sqrt{5} \sqrt{35} + \sqrt{5} \sqrt{34} - \sqrt{7} (\sqrt{35} + \sqrt{34})$

Step 3.3

Apply the distributive property.

$\sqrt{5} \sqrt{35} + \sqrt{5} \sqrt{34} - \sqrt{7} \sqrt{35} - \sqrt{7} \sqrt{34}$

Step 4

Simplify each term.

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Step 4.1

Combine using the product rule for radicals.

$\sqrt{5 \cdot 35} + \sqrt{5} \sqrt{34} - \sqrt{7} \sqrt{35} - \sqrt{7} \sqrt{34}$

Step 4.2

Multiply $5$ by $35$ .

$\sqrt{175} + \sqrt{5} \sqrt{34} - \sqrt{7} \sqrt{35} - \sqrt{7} \sqrt{34}$

Step 4.3

Rewrite $175$ as $5^{2} \cdot 7$ .

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Step 4.3.1

Factor $25$ out of $175$ .

$\sqrt{25 (7)} + \sqrt{5} \sqrt{34} - \sqrt{7} \sqrt{35} - \sqrt{7} \sqrt{34}$

Step 4.3.2

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2} \cdot 7} + \sqrt{5} \sqrt{34} - \sqrt{7} \sqrt{35} - \sqrt{7} \sqrt{34}$

Step 4.4

Pull terms out from under the radical.

$5 \sqrt{7} + \sqrt{5} \sqrt{34} - \sqrt{7} \sqrt{35} - \sqrt{7} \sqrt{34}$

Step 4.5

Combine using the product rule for radicals.

$5 \sqrt{7} + \sqrt{5 \cdot 34} - \sqrt{7} \sqrt{35} - \sqrt{7} \sqrt{34}$

Step 4.6

Multiply $5$ by $34$ .

$5 \sqrt{7} + \sqrt{170} - \sqrt{7} \sqrt{35} - \sqrt{7} \sqrt{34}$

Step 4.7

Multiply $- \sqrt{7} \sqrt{35}$ .

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Step 4.7.1

Combine using the product rule for radicals.

$5 \sqrt{7} + \sqrt{170} - \sqrt{35 \cdot 7} - \sqrt{7} \sqrt{34}$

Step 4.7.2

Multiply $35$ by $7$ .

$5 \sqrt{7} + \sqrt{170} - \sqrt{245} - \sqrt{7} \sqrt{34}$

Step 4.8

Rewrite $245$ as $7^{2} \cdot 5$ .

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Step 4.8.1

Factor $49$ out of $245$ .

$5 \sqrt{7} + \sqrt{170} - \sqrt{49 (5)} - \sqrt{7} \sqrt{34}$

Step 4.8.2

Rewrite $49$ as $7^{2}$ .

$5 \sqrt{7} + \sqrt{170} - \sqrt{7^{2} \cdot 5} - \sqrt{7} \sqrt{34}$

Step 4.9

Pull terms out from under the radical.

$5 \sqrt{7} + \sqrt{170} - (7 \sqrt{5}) - \sqrt{7} \sqrt{34}$

Step 4.10

Multiply $7$ by $- 1$ .

$5 \sqrt{7} + \sqrt{170} - 7 \sqrt{5} - \sqrt{7} \sqrt{34}$

Step 4.11

Multiply $- \sqrt{7} \sqrt{34}$ .

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Step 4.11.1

Combine using the product rule for radicals.

$5 \sqrt{7} + \sqrt{170} - 7 \sqrt{5} - \sqrt{34 \cdot 7}$

Step 4.11.2

Multiply $34$ by $7$ .

$5 \sqrt{7} + \sqrt{170} - 7 \sqrt{5} - \sqrt{238}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt{7} + \sqrt{170} - 7 \sqrt{5} - \sqrt{238}$

Decimal Form:

$- 4.81256309 \dots$