Find the Distance Between Two Points (13/2,4/3) , (-2,10)

$(\frac{13}{2}, \frac{4}{3})$ ,

$(- 2, 10)$

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.

$\sqrt{{((- 2) - \frac{13}{2})}^{2} + {(10 - \frac{4}{3})}^{2}}$

Step 3

Simplify.

To write

$- 2$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$\sqrt{{(- 2 \cdot \frac{2}{2} - \frac{13}{2})}^{2} + {(10 - \frac{4}{3})}^{2}}$

Combine

$- 2$ and

$\frac{2}{2}$ .

$\sqrt{{(\frac{- 2 \cdot 2}{2} - \frac{13}{2})}^{2} + {(10 - \frac{4}{3})}^{2}}$

Combine the numerators over the common denominator.

$\sqrt{{(\frac{- 2 \cdot 2 - 13}{2})}^{2} + {(10 - \frac{4}{3})}^{2}}$

Simplify the numerator.

Multiply

$- 2$ by

$2$ .

$\sqrt{{(\frac{- 4 - 13}{2})}^{2} + {(10 - \frac{4}{3})}^{2}}$

Subtract

$13$ from

$- 4$ .

$\sqrt{{(\frac{- 17}{2})}^{2} + {(10 - \frac{4}{3})}^{2}}$

Move the negative in front of the fraction.

$\sqrt{{(- \frac{17}{2})}^{2} + {(10 - \frac{4}{3})}^{2}}$

Use the power rule

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

Apply the product rule to

$- \frac{17}{2}$ .

$\sqrt{{(- 1)}^{2} {(\frac{17}{2})}^{2} + {(10 - \frac{4}{3})}^{2}}$

Apply the product rule to

$\frac{17}{2}$ .

$\sqrt{{(- 1)}^{2} \frac{17^{2}}{2^{2}} + {(10 - \frac{4}{3})}^{2}}$

Raise

$- 1$ to the power of

$2$ .

$\sqrt{1 \frac{17^{2}}{2^{2}} + {(10 - \frac{4}{3})}^{2}}$

Multiply

$\frac{17^{2}}{2^{2}}$ by

$1$ .

$\sqrt{\frac{17^{2}}{2^{2}} + {(10 - \frac{4}{3})}^{2}}$

Raise

$17$ to the power of

$2$ .

$\sqrt{\frac{289}{2^{2}} + {(10 - \frac{4}{3})}^{2}}$

Raise

$2$ to the power of

$2$ .

$\sqrt{\frac{289}{4} + {(10 - \frac{4}{3})}^{2}}$

To write

$10$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$\sqrt{\frac{289}{4} + {(10 \cdot \frac{3}{3} - \frac{4}{3})}^{2}}$

Combine

$10$ and

$\frac{3}{3}$ .

$\sqrt{\frac{289}{4} + {(\frac{10 \cdot 3}{3} - \frac{4}{3})}^{2}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{289}{4} + {(\frac{10 \cdot 3 - 4}{3})}^{2}}$

Simplify the numerator.

Multiply

$10$ by

$3$ .

$\sqrt{\frac{289}{4} + {(\frac{30 - 4}{3})}^{2}}$

Subtract

$4$ from

$30$ .

$\sqrt{\frac{289}{4} + {(\frac{26}{3})}^{2}}$

Apply the product rule to

$\frac{26}{3}$ .

$\sqrt{\frac{289}{4} + \frac{26^{2}}{3^{2}}}$

Raise

$26$ to the power of

$2$ .

$\sqrt{\frac{289}{4} + \frac{676}{3^{2}}}$

Raise

$3$ to the power of

$2$ .

$\sqrt{\frac{289}{4} + \frac{676}{9}}$

To write

$\frac{289}{4}$ as a fraction with a common denominator, multiply by

$\frac{9}{9}$ .

$\sqrt{\frac{289}{4} \cdot \frac{9}{9} + \frac{676}{9}}$

To write

$\frac{676}{9}$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$\sqrt{\frac{289}{4} \cdot \frac{9}{9} + \frac{676}{9} \cdot \frac{4}{4}}$

Write each expression with a common denominator of

$36$ , by multiplying each by an appropriate factor of

$1$ .

Multiply

$\frac{289}{4}$ by

$\frac{9}{9}$ .

$\sqrt{\frac{289 \cdot 9}{4 \cdot 9} + \frac{676}{9} \cdot \frac{4}{4}}$

Multiply

$4$ by

$9$ .

$\sqrt{\frac{289 \cdot 9}{36} + \frac{676}{9} \cdot \frac{4}{4}}$

Multiply

$\frac{676}{9}$ by

$\frac{4}{4}$ .

$\sqrt{\frac{289 \cdot 9}{36} + \frac{676 \cdot 4}{9 \cdot 4}}$

Multiply

$9$ by

$4$ .

$\sqrt{\frac{289 \cdot 9}{36} + \frac{676 \cdot 4}{36}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{289 \cdot 9 + 676 \cdot 4}{36}}$

Simplify the numerator.

Multiply

$289$ by

$9$ .

$\sqrt{\frac{2601 + 676 \cdot 4}{36}}$

Multiply

$676$ by

$4$ .

$\sqrt{\frac{2601 + 2704}{36}}$

Add

$2601$ and

$2704$ .

$\sqrt{\frac{5305}{36}}$

Rewrite

$\sqrt{\frac{5305}{36}}$ as

$\frac{\sqrt{5305}}{\sqrt{36}}$ .

$\frac{\sqrt{5305}}{\sqrt{36}}$

Simplify the denominator.

Rewrite

$36$ as

$6^{2}$ .

$\frac{\sqrt{5305}}{\sqrt{6^{2}}}$

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

$\frac{\sqrt{5305}}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{5305}}{6}$

Decimal Form:

$12.13923848 \dots$

Step 5

Find the Distance Between Two Points (13/2,4/3) , (-2,10)

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