Trigonometry Examples

$\sqrt{9^{2} - {(- \sqrt{3})}^{2}}$

Step 1

Raise $9$ to the power of $2$ .

$\sqrt{81 - {(- \sqrt{3})}^{2}}$

Step 2

Apply the product rule to $- \sqrt{3}$ .

$\sqrt{81 - ({(- 1)}^{2} {\sqrt{3}}^{2})}$

Step 3

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.1

Move ${(- 1)}^{2}$ .

$\sqrt{81 + {(- 1)}^{2} \cdot - 1 {\sqrt{3}}^{2}}$

Step 3.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 3.2.1

Raise $- 1$ to the power of $1$ .

$\sqrt{81 + {(- 1)}^{2} \cdot {(- 1)}^{1} {\sqrt{3}}^{2}}$

Step 3.2.2

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

$\sqrt{81 + {(- 1)}^{2 + 1} {\sqrt{3}}^{2}}$

Step 3.3

Add $2$ and $1$ .

$\sqrt{81 + {(- 1)}^{3} {\sqrt{3}}^{2}}$

Step 4

Raise $- 1$ to the power of $3$ .

$\sqrt{81 - {\sqrt{3}}^{2}}$

Step 5

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

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Step 5.1

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

$\sqrt{81 - {(3^{\frac{1}{2}})}^{2}}$

Step 5.2

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

$\sqrt{81 - 3^{\frac{1}{2} \cdot 2}}$

Step 5.3

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

$\sqrt{81 - 3^{\frac{2}{2}}}$

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

$\sqrt{81 - 3^{\frac{2}{2}}}$

Step 5.4.2

Rewrite the expression.

$\sqrt{81 - 3^{1}}$

Step 5.5

Evaluate the exponent.

$\sqrt{81 - 1 \cdot 3}$

Step 6

Simplify the expression.

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Step 6.1

Multiply $- 1$ by $3$ .

$\sqrt{81 - 3}$

Step 6.2

Subtract $3$ from $81$ .

$\sqrt{78}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\sqrt{78}$

Decimal Form:

$8.83176086 \dots$