Precalculus Examples

$(\frac{8}{7}, \frac{7}{8})$ , $(\frac{2}{7}, \frac{7}{8})$

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.

$\sqrt{{(\frac{2}{7} - \frac{8}{7})}^{2} + {(\frac{7}{8} - \frac{7}{8})}^{2}}$

Step 3

Simplify.

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Step 3.1

Combine the numerators over the common denominator.

$\sqrt{{(\frac{2 - 8}{7})}^{2} + {(\frac{7}{8} - \frac{7}{8})}^{2}}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Subtract $8$ from $2$ .

$\sqrt{{(\frac{- 6}{7})}^{2} + {(\frac{7}{8} - \frac{7}{8})}^{2}}$

Step 3.2.2

Move the negative in front of the fraction.

$\sqrt{{(- \frac{6}{7})}^{2} + {(\frac{7}{8} - \frac{7}{8})}^{2}}$

Step 3.3

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

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Step 3.3.1

Apply the product rule to $- \frac{6}{7}$ .

$\sqrt{{(- 1)}^{2} {(\frac{6}{7})}^{2} + {(\frac{7}{8} - \frac{7}{8})}^{2}}$

Step 3.3.2

Apply the product rule to $\frac{6}{7}$ .

$\sqrt{{(- 1)}^{2} \frac{6^{2}}{7^{2}} + {(\frac{7}{8} - \frac{7}{8})}^{2}}$

Step 3.4

Simplify the expression.

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Step 3.4.1

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

$\sqrt{1 \frac{6^{2}}{7^{2}} + {(\frac{7}{8} - \frac{7}{8})}^{2}}$

Step 3.4.2

Multiply $\frac{6^{2}}{7^{2}}$ by $1$ .

$\sqrt{\frac{6^{2}}{7^{2}} + {(\frac{7}{8} - \frac{7}{8})}^{2}}$

Step 3.4.3

Raise $6$ to the power of $2$ .

$\sqrt{\frac{36}{7^{2}} + {(\frac{7}{8} - \frac{7}{8})}^{2}}$

Step 3.4.4

Raise $7$ to the power of $2$ .

$\sqrt{\frac{36}{49} + {(\frac{7}{8} - \frac{7}{8})}^{2}}$

Step 3.5

Combine the numerators over the common denominator.

$\sqrt{\frac{36}{49} + {(\frac{7 - 7}{8})}^{2}}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Subtract $7$ from $7$ .

$\sqrt{\frac{36}{49} + {(\frac{0}{8})}^{2}}$

Step 3.6.2

Divide $0$ by $8$ .

$\sqrt{\frac{36}{49} + 0^{2}}$

Step 3.6.3

Raising $0$ to any positive power yields $0$ .

$\sqrt{\frac{36}{49} + 0}$

Step 3.6.4

Add $\frac{36}{49}$ and $0$ .

$\sqrt{\frac{36}{49}}$

Step 3.7

Rewrite $\sqrt{\frac{36}{49}}$ as $\frac{\sqrt{36}}{\sqrt{49}}$ .

$\frac{\sqrt{36}}{\sqrt{49}}$

Step 3.8

Simplify the numerator.

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Step 3.8.1

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

$\frac{\sqrt{6^{2}}}{\sqrt{49}}$

Step 3.8.2

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

$\frac{6}{\sqrt{49}}$

Step 3.9

Simplify the denominator.

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Step 3.9.1

Rewrite $49$ as $7^{2}$ .

$\frac{6}{\sqrt{7^{2}}}$

Step 3.9.2

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

$\frac{6}{7}$

Step 4