Precalculus Examples

$(\frac{3}{2}, \frac{2}{3})$ , $(1, - 1)$

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.

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

Step 3

Simplify.

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Step 3.1

Write $1$ as a fraction with a common denominator.

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 3.3

Subtract $3$ from $2$ .

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

Step 3.4

Move the negative in front of the fraction.

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

Step 3.5

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

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Step 3.5.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 3.5.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 3.6

Raise $- 1$ to the power of $2$ .

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

Step 3.7

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

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

Step 3.8

One to any power is one.

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

Step 3.9

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

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

Step 3.10

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3.11

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 3.12

Combine the numerators over the common denominator.

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

Step 3.13

Simplify the numerator.

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Step 3.13.1

Multiply $- 1$ by $3$ .

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

Step 3.13.2

Subtract $2$ from $- 3$ .

$\sqrt{\frac{1}{4} + {(\frac{- 5}{3})}^{2}}$

Step 3.14

Move the negative in front of the fraction.

$\sqrt{\frac{1}{4} + {(- \frac{5}{3})}^{2}}$

Step 3.15

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

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Step 3.15.1

Apply the product rule to $- \frac{5}{3}$ .

$\sqrt{\frac{1}{4} + {(- 1)}^{2} {(\frac{5}{3})}^{2}}$

Step 3.15.2

Apply the product rule to $\frac{5}{3}$ .

$\sqrt{\frac{1}{4} + {(- 1)}^{2} \frac{5^{2}}{3^{2}}}$

Step 3.16

Raise $- 1$ to the power of $2$ .

$\sqrt{\frac{1}{4} + 1 \frac{5^{2}}{3^{2}}}$

Step 3.17

Multiply $\frac{5^{2}}{3^{2}}$ by $1$ .

$\sqrt{\frac{1}{4} + \frac{5^{2}}{3^{2}}}$

Step 3.18

Raise $5$ to the power of $2$ .

$\sqrt{\frac{1}{4} + \frac{25}{3^{2}}}$

Step 3.19

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

$\sqrt{\frac{1}{4} + \frac{25}{9}}$

Step 3.20

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\sqrt{\frac{1}{4} \cdot \frac{9}{9} + \frac{25}{9}}$

Step 3.21

To write $\frac{25}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{1}{4} \cdot \frac{9}{9} + \frac{25}{9} \cdot \frac{4}{4}}$

Step 3.22

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.22.1

Multiply $\frac{1}{4}$ by $\frac{9}{9}$ .

$\sqrt{\frac{9}{4 \cdot 9} + \frac{25}{9} \cdot \frac{4}{4}}$

Step 3.22.2

Multiply $4$ by $9$ .

$\sqrt{\frac{9}{36} + \frac{25}{9} \cdot \frac{4}{4}}$

Step 3.22.3

Multiply $\frac{25}{9}$ by $\frac{4}{4}$ .

$\sqrt{\frac{9}{36} + \frac{25 \cdot 4}{9 \cdot 4}}$

Step 3.22.4

Multiply $9$ by $4$ .

$\sqrt{\frac{9}{36} + \frac{25 \cdot 4}{36}}$

Step 3.23

Combine the numerators over the common denominator.

$\sqrt{\frac{9 + 25 \cdot 4}{36}}$

Step 3.24

Simplify the numerator.

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Step 3.24.1

Multiply $25$ by $4$ .

$\sqrt{\frac{9 + 100}{36}}$

Step 3.24.2

Add $9$ and $100$ .

$\sqrt{\frac{109}{36}}$

Step 3.25

Rewrite $\sqrt{\frac{109}{36}}$ as $\frac{\sqrt{109}}{\sqrt{36}}$ .

$\frac{\sqrt{109}}{\sqrt{36}}$

Step 3.26

Simplify the denominator.

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Step 3.26.1

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{109}}{\sqrt{6^{2}}}$

Step 3.26.2

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

$\frac{\sqrt{109}}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{109}}{6}$

Decimal Form:

$1.74005108 \dots$

Step 5