Trigonometry Examples

$\sqrt{69^{2} - {(\frac{21 \sqrt{3}}{2})}^{2}}$

Step 1

Raise $69$ to the power of $2$ .

$\sqrt{4761 - {(\frac{21 \sqrt{3}}{2})}^{2}}$

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1

Apply the product rule to $\frac{21 \sqrt{3}}{2}$ .

$\sqrt{4761 - \frac{{(21 \sqrt{3})}^{2}}{2^{2}}}$

Step 2.2

Apply the product rule to $21 \sqrt{3}$ .

$\sqrt{4761 - \frac{21^{2} {\sqrt{3}}^{2}}{2^{2}}}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $21$ to the power of $2$ .

$\sqrt{4761 - \frac{441 {\sqrt{3}}^{2}}{2^{2}}}$

Step 3.2

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

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Step 3.2.1

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

$\sqrt{4761 - \frac{441 {(3^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 3.2.2

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

$\sqrt{4761 - \frac{441 \cdot 3^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 3.2.3

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

$\sqrt{4761 - \frac{441 \cdot 3^{\frac{2}{2}}}{2^{2}}}$

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

$\sqrt{4761 - \frac{441 \cdot 3^{\frac{2}{2}}}{2^{2}}}$

Step 3.2.4.2

Rewrite the expression.

$\sqrt{4761 - \frac{441 \cdot 3^{1}}{2^{2}}}$

Step 3.2.5

Evaluate the exponent.

$\sqrt{4761 - \frac{441 \cdot 3}{2^{2}}}$

Step 4

Simplify the expression.

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Step 4.1

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

$\sqrt{4761 - \frac{441 \cdot 3}{4}}$

Step 4.2

Multiply $441$ by $3$ .

$\sqrt{4761 - \frac{1323}{4}}$

Step 5

To write $4761$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{4761 \cdot \frac{4}{4} - \frac{1323}{4}}$

Step 6

Combine $4761$ and $\frac{4}{4}$ .

$\sqrt{\frac{4761 \cdot 4}{4} - \frac{1323}{4}}$

Step 7

Combine the numerators over the common denominator.

$\sqrt{\frac{4761 \cdot 4 - 1323}{4}}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $4761$ by $4$ .

$\sqrt{\frac{19044 - 1323}{4}}$

Step 8.2

Subtract $1323$ from $19044$ .

$\sqrt{\frac{17721}{4}}$

Step 9

Rewrite $\sqrt{\frac{17721}{4}}$ as $\frac{\sqrt{17721}}{\sqrt{4}}$ .

$\frac{\sqrt{17721}}{\sqrt{4}}$

Step 10

Simplify the numerator.

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Step 10.1

Rewrite $17721$ as $3^{2} \cdot 1969$ .

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Step 10.1.1

Factor $9$ out of $17721$ .

$\frac{\sqrt{9 (1969)}}{\sqrt{4}}$

Step 10.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 1969}}{\sqrt{4}}$

Step 10.2

Pull terms out from under the radical.

$\frac{3 \sqrt{1969}}{\sqrt{4}}$

Step 11

Simplify the denominator.

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Step 11.1

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

$\frac{3 \sqrt{1969}}{\sqrt{2^{2}}}$

Step 11.2

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

$\frac{3 \sqrt{1969}}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{1969}}{2}$

Decimal Form:

$66.56012319 \dots$