Basic Math Examples

$(6 \sqrt{11} + 5 \sqrt{3}) (2 \sqrt{11} - \sqrt{3})$

Step 1

Expand $(6 \sqrt{11} + 5 \sqrt{3}) (2 \sqrt{11} - \sqrt{3})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$6 \sqrt{11} (2 \sqrt{11} - \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11} - \sqrt{3})$

Step 1.2

Apply the distributive property.

$6 \sqrt{11} (2 \sqrt{11}) + 6 \sqrt{11} (- \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11} - \sqrt{3})$

Step 1.3

Apply the distributive property.

$6 \sqrt{11} (2 \sqrt{11}) + 6 \sqrt{11} (- \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

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 $6 \sqrt{11} (2 \sqrt{11})$ .

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Step 2.1.1.1

Multiply $2$ by $6$ .

$12 \sqrt{11} \sqrt{11} + 6 \sqrt{11} (- \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.1.2

Raise $\sqrt{11}$ to the power of $1$ .

$12 ({\sqrt{11}}^{1} \sqrt{11}) + 6 \sqrt{11} (- \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.1.3

Raise $\sqrt{11}$ to the power of $1$ .

$12 ({\sqrt{11}}^{1} {\sqrt{11}}^{1}) + 6 \sqrt{11} (- \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.1.4

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

$12 {\sqrt{11}}^{1 + 1} + 6 \sqrt{11} (- \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.1.5

Add $1$ and $1$ .

$12 {\sqrt{11}}^{2} + 6 \sqrt{11} (- \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.2

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$12 {(11^{\frac{1}{2}})}^{2} + 6 \sqrt{11} (- \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.2.2

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

$12 \cdot 11^{\frac{1}{2} \cdot 2} + 6 \sqrt{11} (- \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.2.3

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

$12 \cdot 11^{\frac{2}{2}} + 6 \sqrt{11} (- \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.4.1

Cancel the common factor.

$12 \cdot 11^{\frac{2}{2}} + 6 \sqrt{11} (- \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.2.4.2

Rewrite the expression.

$12 \cdot 11^{1} + 6 \sqrt{11} (- \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.2.5

Evaluate the exponent.

$12 \cdot 11 + 6 \sqrt{11} (- \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.3

Multiply $12$ by $11$ .

$132 + 6 \sqrt{11} (- \sqrt{3}) + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.4

Multiply $6 \sqrt{11} (- \sqrt{3})$ .

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Step 2.1.4.1

Multiply $- 1$ by $6$ .

$132 - 6 \sqrt{11} \sqrt{3} + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.4.2

Combine using the product rule for radicals.

$132 - 6 \sqrt{3 \cdot 11} + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.4.3

Multiply $3$ by $11$ .

$132 - 6 \sqrt{33} + 5 \sqrt{3} (2 \sqrt{11}) + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.5

Multiply $5 \sqrt{3} (2 \sqrt{11})$ .

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Step 2.1.5.1

Multiply $2$ by $5$ .

$132 - 6 \sqrt{33} + 10 \sqrt{3} \sqrt{11} + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.5.2

Combine using the product rule for radicals.

$132 - 6 \sqrt{33} + 10 \sqrt{11 \cdot 3} + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.5.3

Multiply $11$ by $3$ .

$132 - 6 \sqrt{33} + 10 \sqrt{33} + 5 \sqrt{3} (- \sqrt{3})$

Step 2.1.6

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

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Step 2.1.6.1

Multiply $- 1$ by $5$ .

$132 - 6 \sqrt{33} + 10 \sqrt{33} - 5 \sqrt{3} \sqrt{3}$

Step 2.1.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$132 - 6 \sqrt{33} + 10 \sqrt{33} - 5 ({\sqrt{3}}^{1} \sqrt{3})$

Step 2.1.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$132 - 6 \sqrt{33} + 10 \sqrt{33} - 5 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 2.1.6.4

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

$132 - 6 \sqrt{33} + 10 \sqrt{33} - 5 {\sqrt{3}}^{1 + 1}$

Step 2.1.6.5

Add $1$ and $1$ .

$132 - 6 \sqrt{33} + 10 \sqrt{33} - 5 {\sqrt{3}}^{2}$

Step 2.1.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$132 - 6 \sqrt{33} + 10 \sqrt{33} - 5 {(3^{\frac{1}{2}})}^{2}$

Step 2.1.7.2

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

$132 - 6 \sqrt{33} + 10 \sqrt{33} - 5 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 2.1.7.3

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

$132 - 6 \sqrt{33} + 10 \sqrt{33} - 5 \cdot 3^{\frac{2}{2}}$

Step 2.1.7.4

Cancel the common factor of $2$ .

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Step 2.1.7.4.1

Cancel the common factor.

$132 - 6 \sqrt{33} + 10 \sqrt{33} - 5 \cdot 3^{\frac{2}{2}}$

Step 2.1.7.4.2

Rewrite the expression.

$132 - 6 \sqrt{33} + 10 \sqrt{33} - 5 \cdot 3^{1}$

Step 2.1.7.5

Evaluate the exponent.

$132 - 6 \sqrt{33} + 10 \sqrt{33} - 5 \cdot 3$

Step 2.1.8

Multiply $- 5$ by $3$ .

$132 - 6 \sqrt{33} + 10 \sqrt{33} - 15$

Step 2.2

Subtract $15$ from $132$ .

$117 - 6 \sqrt{33} + 10 \sqrt{33}$

Step 2.3

Add $- 6 \sqrt{33}$ and $10 \sqrt{33}$ .

$117 + 4 \sqrt{33}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$117 + 4 \sqrt{33}$

Decimal Form:

$139.97825058 \dots$