Algebra Examples

${(5 + \sqrt{12})}^{2}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.1.1.1

Factor $4$ out of $12$ .

${(5 + \sqrt{4 (3)})}^{2}$

Step 1.1.1.2

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

${(5 + \sqrt{2^{2} \cdot 3})}^{2}$

Step 1.1.2

Pull terms out from under the radical.

${(5 + 2 \sqrt{3})}^{2}$

Step 1.2

Rewrite ${(5 + 2 \sqrt{3})}^{2}$ as $(5 + 2 \sqrt{3}) (5 + 2 \sqrt{3})$ .

$(5 + 2 \sqrt{3}) (5 + 2 \sqrt{3})$

Step 2

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

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Step 2.1

Apply the distributive property.

$5 (5 + 2 \sqrt{3}) + 2 \sqrt{3} (5 + 2 \sqrt{3})$

Step 2.2

Apply the distributive property.

$5 \cdot 5 + 5 (2 \sqrt{3}) + 2 \sqrt{3} (5 + 2 \sqrt{3})$

Step 2.3

Apply the distributive property.

$5 \cdot 5 + 5 (2 \sqrt{3}) + 2 \sqrt{3} \cdot 5 + 2 \sqrt{3} (2 \sqrt{3})$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $5$ by $5$ .

$25 + 5 (2 \sqrt{3}) + 2 \sqrt{3} \cdot 5 + 2 \sqrt{3} (2 \sqrt{3})$

Step 3.1.2

Multiply $2$ by $5$ .

$25 + 10 \sqrt{3} + 2 \sqrt{3} \cdot 5 + 2 \sqrt{3} (2 \sqrt{3})$

Step 3.1.3

Multiply $5$ by $2$ .

$25 + 10 \sqrt{3} + 10 \sqrt{3} + 2 \sqrt{3} (2 \sqrt{3})$

Step 3.1.4

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

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Step 3.1.4.1

Multiply $2$ by $2$ .

$25 + 10 \sqrt{3} + 10 \sqrt{3} + 4 \sqrt{3} \sqrt{3}$

Step 3.1.4.2

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

$25 + 10 \sqrt{3} + 10 \sqrt{3} + 4 ({\sqrt{3}}^{1} \sqrt{3})$

Step 3.1.4.3

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

$25 + 10 \sqrt{3} + 10 \sqrt{3} + 4 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 3.1.4.4

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

$25 + 10 \sqrt{3} + 10 \sqrt{3} + 4 {\sqrt{3}}^{1 + 1}$

Step 3.1.4.5

Add $1$ and $1$ .

$25 + 10 \sqrt{3} + 10 \sqrt{3} + 4 {\sqrt{3}}^{2}$

Step 3.1.5

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

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Step 3.1.5.1

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

$25 + 10 \sqrt{3} + 10 \sqrt{3} + 4 {(3^{\frac{1}{2}})}^{2}$

Step 3.1.5.2

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

$25 + 10 \sqrt{3} + 10 \sqrt{3} + 4 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 3.1.5.3

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

$25 + 10 \sqrt{3} + 10 \sqrt{3} + 4 \cdot 3^{\frac{2}{2}}$

Step 3.1.5.4

Cancel the common factor of $2$ .

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Step 3.1.5.4.1

Cancel the common factor.

$25 + 10 \sqrt{3} + 10 \sqrt{3} + 4 \cdot 3^{\frac{2}{2}}$

Step 3.1.5.4.2

Rewrite the expression.

$25 + 10 \sqrt{3} + 10 \sqrt{3} + 4 \cdot 3^{1}$

Step 3.1.5.5

Evaluate the exponent.

$25 + 10 \sqrt{3} + 10 \sqrt{3} + 4 \cdot 3$

Step 3.1.6

Multiply $4$ by $3$ .

$25 + 10 \sqrt{3} + 10 \sqrt{3} + 12$

Step 3.2

Add $25$ and $12$ .

$37 + 10 \sqrt{3} + 10 \sqrt{3}$

Step 3.3

Add $10 \sqrt{3}$ and $10 \sqrt{3}$ .

$37 + 20 \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$37 + 20 \sqrt{3}$

Decimal Form:

$71.64101615 \dots$