Precalculus Examples

$(2, \frac{π}{3})$ , $(2, \frac{11 π}{6})$

Step 1

Use the distance formula to determine the distance between the two points.

$Distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Step 2

Substitute the actual values of the points into the distance formula.

$\sqrt{{(2 - 2)}^{2} + {(\frac{11 π}{6} - \frac{π}{3})}^{2}}$

Step 3

Simplify.

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Step 3.1

Subtract $2$ from $2$ .

$\sqrt{0^{2} + {(\frac{11 π}{6} - \frac{π}{3})}^{2}}$

Step 3.2

Raising $0$ to any positive power yields $0$ .

$\sqrt{0 + {(\frac{11 π}{6} - \frac{π}{3})}^{2}}$

Step 3.3

To write $- \frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{0 + {(\frac{11 π}{6} - \frac{π}{3} \cdot \frac{2}{2})}^{2}}$

Step 3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

$\sqrt{0 + {(\frac{11 π}{6} - \frac{π \cdot 2}{3 \cdot 2})}^{2}}$

Step 3.4.2

Multiply $3$ by $2$ .

$\sqrt{0 + {(\frac{11 π}{6} - \frac{π \cdot 2}{6})}^{2}}$

Step 3.5

Combine the numerators over the common denominator.

$\sqrt{0 + {(\frac{11 π - π \cdot 2}{6})}^{2}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $2$ by $- 1$ .

$\sqrt{0 + {(\frac{11 π - 2 π}{6})}^{2}}$

Step 3.6.2

Subtract $2 π$ from $11 π$ .

$\sqrt{0 + {(\frac{9 π}{6})}^{2}}$

Step 3.7

Cancel the common factor of $9$ and $6$ .

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Step 3.7.1

Factor $3$ out of $9 π$ .

$\sqrt{0 + {(\frac{3 (3 π)}{6})}^{2}}$

Step 3.7.2

Cancel the common factors.

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Step 3.7.2.1

Factor $3$ out of $6$ .

$\sqrt{0 + {(\frac{3 (3 π)}{3 (2)})}^{2}}$

Step 3.7.2.2

Cancel the common factor.

$\sqrt{0 + {(\frac{3 (3 π)}{3 \cdot 2})}^{2}}$

Step 3.7.2.3

Rewrite the expression.

$\sqrt{0 + {(\frac{3 π}{2})}^{2}}$

Step 3.8

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

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Step 3.8.1

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

$\sqrt{0 + \frac{{(3 π)}^{2}}{2^{2}}}$

Step 3.8.2

Apply the product rule to $3 π$ .

$\sqrt{0 + \frac{3^{2} π^{2}}{2^{2}}}$

Step 3.9

Raise $3$ to the power of $2$ .

$\sqrt{0 + \frac{9 π^{2}}{2^{2}}}$

Step 3.10

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

$\sqrt{0 + \frac{9 π^{2}}{4}}$

Step 3.11

Add $0$ and $\frac{9 π^{2}}{4}$ .

$\sqrt{\frac{9 π^{2}}{4}}$

Step 3.12

Rewrite $\sqrt{\frac{9 π^{2}}{4}}$ as $\frac{\sqrt{9 π^{2}}}{\sqrt{4}}$ .

$\frac{\sqrt{9 π^{2}}}{\sqrt{4}}$

Step 3.13

Simplify the numerator.

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Step 3.13.1

Rewrite $9 π^{2}$ as ${(3 π)}^{2}$ .

$\frac{\sqrt{{(3 π)}^{2}}}{\sqrt{4}}$

Step 3.13.2

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

$\frac{3 π}{\sqrt{4}}$

Step 3.14

Simplify the denominator.

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Step 3.14.1

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

$\frac{3 π}{\sqrt{2^{2}}}$

Step 3.14.2

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

$\frac{3 π}{2}$

Step 4