Algebra Examples

$(\sqrt{7} - \sqrt{10}) (\sqrt{5} + \sqrt{14})$

Step 1

Expand $(\sqrt{7} - \sqrt{10}) (\sqrt{5} + \sqrt{14})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{7} (\sqrt{5} + \sqrt{14}) - \sqrt{10} (\sqrt{5} + \sqrt{14})$

Step 1.2

Apply the distributive property.

$\sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{14} - \sqrt{10} (\sqrt{5} + \sqrt{14})$

Step 1.3

Apply the distributive property.

$\sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{14} - \sqrt{10} \sqrt{5} - \sqrt{10} \sqrt{14}$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine using the product rule for radicals.

$\sqrt{7 \cdot 5} + \sqrt{7} \sqrt{14} - \sqrt{10} \sqrt{5} - \sqrt{10} \sqrt{14}$

Step 2.1.2

Multiply $7$ by $5$ .

$\sqrt{35} + \sqrt{7} \sqrt{14} - \sqrt{10} \sqrt{5} - \sqrt{10} \sqrt{14}$

Step 2.1.3

Combine using the product rule for radicals.

$\sqrt{35} + \sqrt{7 \cdot 14} - \sqrt{10} \sqrt{5} - \sqrt{10} \sqrt{14}$

Step 2.1.4

Multiply $7$ by $14$ .

$\sqrt{35} + \sqrt{98} - \sqrt{10} \sqrt{5} - \sqrt{10} \sqrt{14}$

Step 2.1.5

Rewrite $98$ as $7^{2} \cdot 2$ .

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Step 2.1.5.1

Factor $49$ out of $98$ .

$\sqrt{35} + \sqrt{49 (2)} - \sqrt{10} \sqrt{5} - \sqrt{10} \sqrt{14}$

Step 2.1.5.2

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

$\sqrt{35} + \sqrt{7^{2} \cdot 2} - \sqrt{10} \sqrt{5} - \sqrt{10} \sqrt{14}$

Step 2.1.6

Pull terms out from under the radical.

$\sqrt{35} + 7 \sqrt{2} - \sqrt{10} \sqrt{5} - \sqrt{10} \sqrt{14}$

Step 2.1.7

Multiply $- \sqrt{10} \sqrt{5}$ .

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Step 2.1.7.1

Combine using the product rule for radicals.

$\sqrt{35} + 7 \sqrt{2} - \sqrt{5 \cdot 10} - \sqrt{10} \sqrt{14}$

Step 2.1.7.2

Multiply $5$ by $10$ .

$\sqrt{35} + 7 \sqrt{2} - \sqrt{50} - \sqrt{10} \sqrt{14}$

Step 2.1.8

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 2.1.8.1

Factor $25$ out of $50$ .

$\sqrt{35} + 7 \sqrt{2} - \sqrt{25 (2)} - \sqrt{10} \sqrt{14}$

Step 2.1.8.2

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

$\sqrt{35} + 7 \sqrt{2} - \sqrt{5^{2} \cdot 2} - \sqrt{10} \sqrt{14}$

Step 2.1.9

Pull terms out from under the radical.

$\sqrt{35} + 7 \sqrt{2} - (5 \sqrt{2}) - \sqrt{10} \sqrt{14}$

Step 2.1.10

Multiply $5$ by $- 1$ .

$\sqrt{35} + 7 \sqrt{2} - 5 \sqrt{2} - \sqrt{10} \sqrt{14}$

Step 2.1.11

Multiply $- \sqrt{10} \sqrt{14}$ .

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Step 2.1.11.1

Combine using the product rule for radicals.

$\sqrt{35} + 7 \sqrt{2} - 5 \sqrt{2} - \sqrt{14 \cdot 10}$

Step 2.1.11.2

Multiply $14$ by $10$ .

$\sqrt{35} + 7 \sqrt{2} - 5 \sqrt{2} - \sqrt{140}$

Step 2.1.12

Rewrite $140$ as $2^{2} \cdot 35$ .

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Step 2.1.12.1

Factor $4$ out of $140$ .

$\sqrt{35} + 7 \sqrt{2} - 5 \sqrt{2} - \sqrt{4 (35)}$

Step 2.1.12.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.1.13

Pull terms out from under the radical.

$\sqrt{35} + 7 \sqrt{2} - 5 \sqrt{2} - (2 \sqrt{35})$

Step 2.1.14

Multiply $2$ by $- 1$ .

$\sqrt{35} + 7 \sqrt{2} - 5 \sqrt{2} - 2 \sqrt{35}$

Step 2.2

Subtract $2 \sqrt{35}$ from $\sqrt{35}$ .

$- \sqrt{35} + 7 \sqrt{2} - 5 \sqrt{2}$

Step 2.3

Subtract $5 \sqrt{2}$ from $7 \sqrt{2}$ .

$- \sqrt{35} + 2 \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{35} + 2 \sqrt{2}$

Decimal Form:

$- 3.08765265 \dots$