Basic Math Examples

$(9 \sqrt{11} - 6 \sqrt{5}) \cdot (3 \sqrt{22} + 7 \sqrt{15})$

Step 1

Expand $(9 \sqrt{11} - 6 \sqrt{5}) (3 \sqrt{22} + 7 \sqrt{15})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$9 \sqrt{11} (3 \sqrt{22} + 7 \sqrt{15}) - 6 \sqrt{5} (3 \sqrt{22} + 7 \sqrt{15})$

Step 1.2

Apply the distributive property.

$9 \sqrt{11} (3 \sqrt{22}) + 9 \sqrt{11} (7 \sqrt{15}) - 6 \sqrt{5} (3 \sqrt{22} + 7 \sqrt{15})$

Step 1.3

Apply the distributive property.

$9 \sqrt{11} (3 \sqrt{22}) + 9 \sqrt{11} (7 \sqrt{15}) - 6 \sqrt{5} (3 \sqrt{22}) - 6 \sqrt{5} (7 \sqrt{15})$

Step 2

Simplify each term.

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Step 2.1

Multiply $9 \sqrt{11} (3 \sqrt{22})$ .

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Step 2.1.1

Multiply $3$ by $9$ .

$27 \sqrt{11} \sqrt{22} + 9 \sqrt{11} (7 \sqrt{15}) - 6 \sqrt{5} (3 \sqrt{22}) - 6 \sqrt{5} (7 \sqrt{15})$

Step 2.1.2

Combine using the product rule for radicals.

$27 \sqrt{22 \cdot 11} + 9 \sqrt{11} (7 \sqrt{15}) - 6 \sqrt{5} (3 \sqrt{22}) - 6 \sqrt{5} (7 \sqrt{15})$

Step 2.1.3

Multiply $22$ by $11$ .

$27 \sqrt{242} + 9 \sqrt{11} (7 \sqrt{15}) - 6 \sqrt{5} (3 \sqrt{22}) - 6 \sqrt{5} (7 \sqrt{15})$

Step 2.2

Rewrite $242$ as $11^{2} \cdot 2$ .

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Step 2.2.1

Factor $121$ out of $242$ .

$27 \sqrt{121 (2)} + 9 \sqrt{11} (7 \sqrt{15}) - 6 \sqrt{5} (3 \sqrt{22}) - 6 \sqrt{5} (7 \sqrt{15})$

Step 2.2.2

Rewrite $121$ as $11^{2}$ .

$27 \sqrt{11^{2} \cdot 2} + 9 \sqrt{11} (7 \sqrt{15}) - 6 \sqrt{5} (3 \sqrt{22}) - 6 \sqrt{5} (7 \sqrt{15})$

Step 2.3

Pull terms out from under the radical.

$27 (11 \sqrt{2}) + 9 \sqrt{11} (7 \sqrt{15}) - 6 \sqrt{5} (3 \sqrt{22}) - 6 \sqrt{5} (7 \sqrt{15})$

Step 2.4

Multiply $11$ by $27$ .

$297 \sqrt{2} + 9 \sqrt{11} (7 \sqrt{15}) - 6 \sqrt{5} (3 \sqrt{22}) - 6 \sqrt{5} (7 \sqrt{15})$

Step 2.5

Multiply $9 \sqrt{11} (7 \sqrt{15})$ .

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Step 2.5.1

Multiply $7$ by $9$ .

$297 \sqrt{2} + 63 \sqrt{11} \sqrt{15} - 6 \sqrt{5} (3 \sqrt{22}) - 6 \sqrt{5} (7 \sqrt{15})$

Step 2.5.2

Combine using the product rule for radicals.

$297 \sqrt{2} + 63 \sqrt{15 \cdot 11} - 6 \sqrt{5} (3 \sqrt{22}) - 6 \sqrt{5} (7 \sqrt{15})$

Step 2.5.3

Multiply $15$ by $11$ .

$297 \sqrt{2} + 63 \sqrt{165} - 6 \sqrt{5} (3 \sqrt{22}) - 6 \sqrt{5} (7 \sqrt{15})$

Step 2.6

Multiply $- 6 \sqrt{5} (3 \sqrt{22})$ .

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Step 2.6.1

Multiply $3$ by $- 6$ .

$297 \sqrt{2} + 63 \sqrt{165} - 18 \sqrt{5} \sqrt{22} - 6 \sqrt{5} (7 \sqrt{15})$

Step 2.6.2

Combine using the product rule for radicals.

$297 \sqrt{2} + 63 \sqrt{165} - 18 \sqrt{22 \cdot 5} - 6 \sqrt{5} (7 \sqrt{15})$

Step 2.6.3

Multiply $22$ by $5$ .

$297 \sqrt{2} + 63 \sqrt{165} - 18 \sqrt{110} - 6 \sqrt{5} (7 \sqrt{15})$

Step 2.7

Multiply $- 6 \sqrt{5} (7 \sqrt{15})$ .

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Step 2.7.1

Multiply $7$ by $- 6$ .

$297 \sqrt{2} + 63 \sqrt{165} - 18 \sqrt{110} - 42 \sqrt{5} \sqrt{15}$

Step 2.7.2

Combine using the product rule for radicals.

$297 \sqrt{2} + 63 \sqrt{165} - 18 \sqrt{110} - 42 \sqrt{15 \cdot 5}$

Step 2.7.3

Multiply $15$ by $5$ .

$297 \sqrt{2} + 63 \sqrt{165} - 18 \sqrt{110} - 42 \sqrt{75}$

Step 2.8

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 2.8.1

Factor $25$ out of $75$ .

$297 \sqrt{2} + 63 \sqrt{165} - 18 \sqrt{110} - 42 \sqrt{25 (3)}$

Step 2.8.2

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

$297 \sqrt{2} + 63 \sqrt{165} - 18 \sqrt{110} - 42 \sqrt{5^{2} \cdot 3}$

Step 2.9

Pull terms out from under the radical.

$297 \sqrt{2} + 63 \sqrt{165} - 18 \sqrt{110} - 42 (5 \sqrt{3})$

Step 2.10

Multiply $5$ by $- 42$ .

$297 \sqrt{2} + 63 \sqrt{165} - 18 \sqrt{110} - 210 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$297 \sqrt{2} + 63 \sqrt{165} - 18 \sqrt{110} - 210 \sqrt{3}$

Decimal Form:

$676.75481822 \dots$