Basic Math Examples

$(2 \sqrt{15} - \sqrt{14}) (\sqrt{10} + \sqrt{21})$

Step 1

Expand $(2 \sqrt{15} - \sqrt{14}) (\sqrt{10} + \sqrt{21})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$2 \sqrt{15} (\sqrt{10} + \sqrt{21}) - \sqrt{14} (\sqrt{10} + \sqrt{21})$

Step 1.2

Apply the distributive property.

$2 \sqrt{15} \sqrt{10} + 2 \sqrt{15} \sqrt{21} - \sqrt{14} (\sqrt{10} + \sqrt{21})$

Step 1.3

Apply the distributive property.

$2 \sqrt{15} \sqrt{10} + 2 \sqrt{15} \sqrt{21} - \sqrt{14} \sqrt{10} - \sqrt{14} \sqrt{21}$

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

Multiply $2 \sqrt{15} \sqrt{10}$ .

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Step 2.1.1.1

Combine using the product rule for radicals.

$2 \sqrt{10 \cdot 15} + 2 \sqrt{15} \sqrt{21} - \sqrt{14} \sqrt{10} - \sqrt{14} \sqrt{21}$

Step 2.1.1.2

Multiply $10$ by $15$ .

$2 \sqrt{150} + 2 \sqrt{15} \sqrt{21} - \sqrt{14} \sqrt{10} - \sqrt{14} \sqrt{21}$

Step 2.1.2

Rewrite $150$ as $5^{2} \cdot 6$ .

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Step 2.1.2.1

Factor $25$ out of $150$ .

$2 \sqrt{25 (6)} + 2 \sqrt{15} \sqrt{21} - \sqrt{14} \sqrt{10} - \sqrt{14} \sqrt{21}$

Step 2.1.2.2

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

$2 \sqrt{5^{2} \cdot 6} + 2 \sqrt{15} \sqrt{21} - \sqrt{14} \sqrt{10} - \sqrt{14} \sqrt{21}$

Step 2.1.3

Pull terms out from under the radical.

$2 (5 \sqrt{6}) + 2 \sqrt{15} \sqrt{21} - \sqrt{14} \sqrt{10} - \sqrt{14} \sqrt{21}$

Step 2.1.4

Multiply $5$ by $2$ .

$10 \sqrt{6} + 2 \sqrt{15} \sqrt{21} - \sqrt{14} \sqrt{10} - \sqrt{14} \sqrt{21}$

Step 2.1.5

Multiply $2 \sqrt{15} \sqrt{21}$ .

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Step 2.1.5.1

Combine using the product rule for radicals.

$10 \sqrt{6} + 2 \sqrt{21 \cdot 15} - \sqrt{14} \sqrt{10} - \sqrt{14} \sqrt{21}$

Step 2.1.5.2

Multiply $21$ by $15$ .

$10 \sqrt{6} + 2 \sqrt{315} - \sqrt{14} \sqrt{10} - \sqrt{14} \sqrt{21}$

Step 2.1.6

Rewrite $315$ as $3^{2} \cdot 35$ .

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Step 2.1.6.1

Factor $9$ out of $315$ .

$10 \sqrt{6} + 2 \sqrt{9 (35)} - \sqrt{14} \sqrt{10} - \sqrt{14} \sqrt{21}$

Step 2.1.6.2

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

$10 \sqrt{6} + 2 \sqrt{3^{2} \cdot 35} - \sqrt{14} \sqrt{10} - \sqrt{14} \sqrt{21}$

Step 2.1.7

Pull terms out from under the radical.

$10 \sqrt{6} + 2 (3 \sqrt{35}) - \sqrt{14} \sqrt{10} - \sqrt{14} \sqrt{21}$

Step 2.1.8

Multiply $3$ by $2$ .

$10 \sqrt{6} + 6 \sqrt{35} - \sqrt{14} \sqrt{10} - \sqrt{14} \sqrt{21}$

Step 2.1.9

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

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Step 2.1.9.1

Combine using the product rule for radicals.

$10 \sqrt{6} + 6 \sqrt{35} - \sqrt{10 \cdot 14} - \sqrt{14} \sqrt{21}$

Step 2.1.9.2

Multiply $10$ by $14$ .

$10 \sqrt{6} + 6 \sqrt{35} - \sqrt{140} - \sqrt{14} \sqrt{21}$

Step 2.1.10

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

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Step 2.1.10.1

Factor $4$ out of $140$ .

$10 \sqrt{6} + 6 \sqrt{35} - \sqrt{4 (35)} - \sqrt{14} \sqrt{21}$

Step 2.1.10.2

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

$10 \sqrt{6} + 6 \sqrt{35} - \sqrt{2^{2} \cdot 35} - \sqrt{14} \sqrt{21}$

Step 2.1.11

Pull terms out from under the radical.

$10 \sqrt{6} + 6 \sqrt{35} - (2 \sqrt{35}) - \sqrt{14} \sqrt{21}$

Step 2.1.12

Multiply $2$ by $- 1$ .

$10 \sqrt{6} + 6 \sqrt{35} - 2 \sqrt{35} - \sqrt{14} \sqrt{21}$

Step 2.1.13

Multiply $- \sqrt{14} \sqrt{21}$ .

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Step 2.1.13.1

Combine using the product rule for radicals.

$10 \sqrt{6} + 6 \sqrt{35} - 2 \sqrt{35} - \sqrt{21 \cdot 14}$

Step 2.1.13.2

Multiply $21$ by $14$ .

$10 \sqrt{6} + 6 \sqrt{35} - 2 \sqrt{35} - \sqrt{294}$

Step 2.1.14

Rewrite $294$ as $7^{2} \cdot 6$ .

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Step 2.1.14.1

Factor $49$ out of $294$ .

$10 \sqrt{6} + 6 \sqrt{35} - 2 \sqrt{35} - \sqrt{49 (6)}$

Step 2.1.14.2

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

$10 \sqrt{6} + 6 \sqrt{35} - 2 \sqrt{35} - \sqrt{7^{2} \cdot 6}$

Step 2.1.15

Pull terms out from under the radical.

$10 \sqrt{6} + 6 \sqrt{35} - 2 \sqrt{35} - (7 \sqrt{6})$

Step 2.1.16

Multiply $7$ by $- 1$ .

$10 \sqrt{6} + 6 \sqrt{35} - 2 \sqrt{35} - 7 \sqrt{6}$

Step 2.2

Subtract $7 \sqrt{6}$ from $10 \sqrt{6}$ .

$3 \sqrt{6} + 6 \sqrt{35} - 2 \sqrt{35}$

Step 2.3

Subtract $2 \sqrt{35}$ from $6 \sqrt{35}$ .

$3 \sqrt{6} + 4 \sqrt{35}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{6} + 4 \sqrt{35}$

Decimal Form:

$31.01278836 \dots$