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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