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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