Trigonometry Examples

$\sqrt{3} (10 + \sqrt{12})$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $4$ out of $12$ .

$\sqrt{3} (10 + \sqrt{4 (3)})$

Step 1.1.2

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

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Simplify by multiplying through.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Move $10$ to the left of $\sqrt{3}$ .

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

Step 3

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

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Step 3.1

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

$10 \cdot \sqrt{3} + 2 ({\sqrt{3}}^{1} \sqrt{3})$

Step 3.2

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

$10 \cdot \sqrt{3} + 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 3.3

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

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

Step 3.4

Add $1$ and $1$ .

$10 \cdot \sqrt{3} + 2 {\sqrt{3}}^{2}$

Step 4

Simplify each term.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

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

Step 4.1.4.2

Rewrite the expression.

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

Step 4.1.5

Evaluate the exponent.

$10 \sqrt{3} + 2 \cdot 3$

Step 4.2

Multiply $2$ by $3$ .

$10 \sqrt{3} + 6$

Step 5

The result can be shown in multiple forms.

Exact Form:

$10 \sqrt{3} + 6$

Decimal Form:

$23.32050807 \dots$