Trigonometry Examples

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

Step 1

Simplify the expression.

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Step 1.1

Subtract $1$ from $- 1$ .

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

Step 1.2

Raise $- 2$ to the power of $2$ .

$\sqrt{4 + {(1 - \sqrt{3})}^{2}}$

Step 1.3

Rewrite ${(1 - \sqrt{3})}^{2}$ as $(1 - \sqrt{3}) (1 - \sqrt{3})$ .

$\sqrt{4 + (1 - \sqrt{3}) (1 - \sqrt{3})}$

Step 2

Expand $(1 - \sqrt{3}) (1 - \sqrt{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\sqrt{4 + 1 (1 - \sqrt{3}) - \sqrt{3} (1 - \sqrt{3})}$

Step 2.2

Apply the distributive property.

$\sqrt{4 + 1 \cdot 1 + 1 (- \sqrt{3}) - \sqrt{3} (1 - \sqrt{3})}$

Step 2.3

Apply the distributive property.

$\sqrt{4 + 1 \cdot 1 + 1 (- \sqrt{3}) - \sqrt{3} \cdot 1 - \sqrt{3} (- \sqrt{3})}$

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 $1$ by $1$ .

$\sqrt{4 + 1 + 1 (- \sqrt{3}) - \sqrt{3} \cdot 1 - \sqrt{3} (- \sqrt{3})}$

Step 3.1.2

Multiply $- \sqrt{3}$ by $1$ .

$\sqrt{4 + 1 - \sqrt{3} - \sqrt{3} \cdot 1 - \sqrt{3} (- \sqrt{3})}$

Step 3.1.3

Multiply $- 1$ by $1$ .

$\sqrt{4 + 1 - \sqrt{3} - \sqrt{3} - \sqrt{3} (- \sqrt{3})}$

Step 3.1.4

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

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Step 3.1.4.1

Multiply $- 1$ by $- 1$ .

$\sqrt{4 + 1 - \sqrt{3} - \sqrt{3} + 1 \sqrt{3} \sqrt{3}}$

Step 3.1.4.2

Multiply $\sqrt{3}$ by $1$ .

$\sqrt{4 + 1 - \sqrt{3} - \sqrt{3} + \sqrt{3} \sqrt{3}}$

Step 3.1.4.3

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

$\sqrt{4 + 1 - \sqrt{3} - \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}$

Step 3.1.4.4

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

$\sqrt{4 + 1 - \sqrt{3} - \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.1.4.5

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

$\sqrt{4 + 1 - \sqrt{3} - \sqrt{3} + {\sqrt{3}}^{1 + 1}}$

Step 3.1.4.6

Add $1$ and $1$ .

$\sqrt{4 + 1 - \sqrt{3} - \sqrt{3} + {\sqrt{3}}^{2}}$

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 $3^{\frac{1}{2}}$ .

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

Step 3.1.5.2

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

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

Step 3.1.5.3

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

$\sqrt{4 + 1 - \sqrt{3} - \sqrt{3} + 3^{\frac{2}{2}}}$

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.

$\sqrt{4 + 1 - \sqrt{3} - \sqrt{3} + 3^{\frac{2}{2}}}$

Step 3.1.5.4.2

Rewrite the expression.

$\sqrt{4 + 1 - \sqrt{3} - \sqrt{3} + 3^{1}}$

Step 3.1.5.5

Evaluate the exponent.

$\sqrt{4 + 1 - \sqrt{3} - \sqrt{3} + 3}$

Step 3.2

Add $1$ and $3$ .

$\sqrt{4 + 4 - \sqrt{3} - \sqrt{3}}$

Step 3.3

Subtract $\sqrt{3}$ from $- \sqrt{3}$ .

$\sqrt{4 + 4 - 2 \sqrt{3}}$

Step 4

Add $4$ and $4$ .

$\sqrt{8 - 2 \sqrt{3}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\sqrt{8 - 2 \sqrt{3}}$

Decimal Form:

$2.12976486 \dots$