Find the Distance Between Two Points (-4/5,4) , (-1,3/2)

$(- \frac{4}{5}, 4)$ ,

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

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.

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

Step 3

Simplify.

Multiply

$- (- \frac{4}{5})$ .

Multiply

$- 1$ by

$- 1$ .

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

Multiply

$\frac{4}{5}$ by

$1$ .

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

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{5}{5}$ .

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

Combine

$- 1$ and

$\frac{5}{5}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply

$- 1$ by

$5$ .

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

Add

$- 5$ and

$4$ .

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

Move the negative in front of the fraction.

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

Use the power rule

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

Apply the product rule to

$- \frac{1}{5}$ .

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

Apply the product rule to

$\frac{1}{5}$ .

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

Raise

$- 1$ to the power of

$2$ .

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

Multiply

$\frac{1^{2}}{5^{2}}$ by

$1$ .

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

One to any power is one.

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

Raise

$5$ to the power of

$2$ .

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

To write

$- 4$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$\sqrt{\frac{1}{25} + {(\frac{3}{2} - 4 \cdot \frac{2}{2})}^{2}}$

Combine

$- 4$ and

$\frac{2}{2}$ .

$\sqrt{\frac{1}{25} + {(\frac{3}{2} + \frac{- 4 \cdot 2}{2})}^{2}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{1}{25} + {(\frac{3 - 4 \cdot 2}{2})}^{2}}$

Simplify the numerator.

Multiply

$- 4$ by

$2$ .

$\sqrt{\frac{1}{25} + {(\frac{3 - 8}{2})}^{2}}$

Subtract

$8$ from

$3$ .

$\sqrt{\frac{1}{25} + {(\frac{- 5}{2})}^{2}}$

Move the negative in front of the fraction.

$\sqrt{\frac{1}{25} + {(- \frac{5}{2})}^{2}}$

Use the power rule

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

Apply the product rule to

$- \frac{5}{2}$ .

$\sqrt{\frac{1}{25} + {(- 1)}^{2} {(\frac{5}{2})}^{2}}$

Apply the product rule to

$\frac{5}{2}$ .

$\sqrt{\frac{1}{25} + {(- 1)}^{2} \frac{5^{2}}{2^{2}}}$

Raise

$- 1$ to the power of

$2$ .

$\sqrt{\frac{1}{25} + 1 \frac{5^{2}}{2^{2}}}$

Multiply

$\frac{5^{2}}{2^{2}}$ by

$1$ .

$\sqrt{\frac{1}{25} + \frac{5^{2}}{2^{2}}}$

Raise

$5$ to the power of

$2$ .

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

Raise

$2$ to the power of

$2$ .

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

To write

$\frac{1}{25}$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

To write

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

$\frac{25}{25}$ .

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

Write each expression with a common denominator of

$100$ , by multiplying each by an appropriate factor of

$1$ .

Multiply

$\frac{1}{25}$ by

$\frac{4}{4}$ .

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

Multiply

$25$ by

$4$ .

$\sqrt{\frac{4}{100} + \frac{25}{4} \cdot \frac{25}{25}}$

Multiply

$\frac{25}{4}$ by

$\frac{25}{25}$ .

$\sqrt{\frac{4}{100} + \frac{25 \cdot 25}{4 \cdot 25}}$

Multiply

$4$ by

$25$ .

$\sqrt{\frac{4}{100} + \frac{25 \cdot 25}{100}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{4 + 25 \cdot 25}{100}}$

Simplify the numerator.

Multiply

$25$ by

$25$ .

$\sqrt{\frac{4 + 625}{100}}$

Add

$4$ and

$625$ .

$\sqrt{\frac{629}{100}}$

Rewrite

$\sqrt{\frac{629}{100}}$ as

$\frac{\sqrt{629}}{\sqrt{100}}$ .

$\frac{\sqrt{629}}{\sqrt{100}}$

Simplify the denominator.

Rewrite

$100$ as

$10^{2}$ .

$\frac{\sqrt{629}}{\sqrt{10^{2}}}$

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

$\frac{\sqrt{629}}{10}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{629}}{10}$

Decimal Form:

$2.50798724 \dots$

Step 5

Find the Distance Between Two Points (-4/5,4) , (-1,3/2)

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