Basic Math Examples

$(3 \sqrt{11} + \sqrt{2}) (3 \sqrt{11} + \sqrt{2})$

Step 1

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

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Step 1.1

Apply the distributive property.

$3 \sqrt{11} (3 \sqrt{11} + \sqrt{2}) + \sqrt{2} (3 \sqrt{11} + \sqrt{2})$

Step 1.2

Apply the distributive property.

$3 \sqrt{11} (3 \sqrt{11}) + 3 \sqrt{11} \sqrt{2} + \sqrt{2} (3 \sqrt{11} + \sqrt{2})$

Step 1.3

Apply the distributive property.

$3 \sqrt{11} (3 \sqrt{11}) + 3 \sqrt{11} \sqrt{2} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

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 $3 \sqrt{11} (3 \sqrt{11})$ .

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Step 2.1.1.1

Multiply $3$ by $3$ .

$9 \sqrt{11} \sqrt{11} + 3 \sqrt{11} \sqrt{2} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

Step 2.1.1.2

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

$9 ({\sqrt{11}}^{1} \sqrt{11}) + 3 \sqrt{11} \sqrt{2} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

Step 2.1.1.3

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

$9 ({\sqrt{11}}^{1} {\sqrt{11}}^{1}) + 3 \sqrt{11} \sqrt{2} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

Step 2.1.1.4

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

$9 {\sqrt{11}}^{1 + 1} + 3 \sqrt{11} \sqrt{2} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

Step 2.1.1.5

Add $1$ and $1$ .

$9 {\sqrt{11}}^{2} + 3 \sqrt{11} \sqrt{2} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

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 $11^{\frac{1}{2}}$ .

$9 {(11^{\frac{1}{2}})}^{2} + 3 \sqrt{11} \sqrt{2} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

Step 2.1.2.2

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

$9 \cdot 11^{\frac{1}{2} \cdot 2} + 3 \sqrt{11} \sqrt{2} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

Step 2.1.2.3

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

$9 \cdot 11^{\frac{2}{2}} + 3 \sqrt{11} \sqrt{2} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

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.

$9 \cdot 11^{\frac{2}{2}} + 3 \sqrt{11} \sqrt{2} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

Step 2.1.2.4.2

Rewrite the expression.

$9 \cdot 11^{1} + 3 \sqrt{11} \sqrt{2} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

Step 2.1.2.5

Evaluate the exponent.

$9 \cdot 11 + 3 \sqrt{11} \sqrt{2} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

Step 2.1.3

Multiply $9$ by $11$ .

$99 + 3 \sqrt{11} \sqrt{2} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

Step 2.1.4

Multiply $3 \sqrt{11} \sqrt{2}$ .

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Step 2.1.4.1

Combine using the product rule for radicals.

$99 + 3 \sqrt{2 \cdot 11} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

Step 2.1.4.2

Multiply $2$ by $11$ .

$99 + 3 \sqrt{22} + \sqrt{2} (3 \sqrt{11}) + \sqrt{2} \sqrt{2}$

Step 2.1.5

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

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Step 2.1.5.1

Combine using the product rule for radicals.

$99 + 3 \sqrt{22} + 3 \sqrt{2 \cdot 11} + \sqrt{2} \sqrt{2}$

Step 2.1.5.2

Multiply $2$ by $11$ .

$99 + 3 \sqrt{22} + 3 \sqrt{22} + \sqrt{2} \sqrt{2}$

Step 2.1.6

Combine using the product rule for radicals.

$99 + 3 \sqrt{22} + 3 \sqrt{22} + \sqrt{2 \cdot 2}$

Step 2.1.7

Multiply $2$ by $2$ .

$99 + 3 \sqrt{22} + 3 \sqrt{22} + \sqrt{4}$

Step 2.1.8

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

$99 + 3 \sqrt{22} + 3 \sqrt{22} + \sqrt{2^{2}}$

Step 2.1.9

Pull terms out from under the radical, assuming positive real numbers.

$99 + 3 \sqrt{22} + 3 \sqrt{22} + 2$

Step 2.2

Add $99$ and $2$ .

$101 + 3 \sqrt{22} + 3 \sqrt{22}$

Step 2.3

Add $3 \sqrt{22}$ and $3 \sqrt{22}$ .

$101 + 6 \sqrt{22}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$101 + 6 \sqrt{22}$

Decimal Form:

$129.14249455 \dots$