Basic Math Examples

$(- \frac{1}{2}, \frac{5}{7})$ , $(- \frac{8}{3}, \frac{1}{21})$

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.

$\sqrt{{((- \frac{8}{3}) - (- \frac{1}{2}))}^{2} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3

Simplify.

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Step 3.1

Multiply $- (- \frac{1}{2})$ .

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Step 3.1.1

Multiply $- 1$ by $- 1$ .

$\sqrt{{(- \frac{8}{3} + 1 (\frac{1}{2}))}^{2} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.1.2

Multiply $\frac{1}{2}$ by $1$ .

$\sqrt{{(- \frac{8}{3} + \frac{1}{2})}^{2} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.2

To write $- \frac{8}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{{(- \frac{8}{3} \cdot \frac{2}{2} + \frac{1}{2})}^{2} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.3

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt{{(- \frac{8}{3} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{3}{3})}^{2} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{8}{3}$ by $\frac{2}{2}$ .

$\sqrt{{(- \frac{8 \cdot 2}{3 \cdot 2} + \frac{1}{2} \cdot \frac{3}{3})}^{2} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.4.2

Multiply $3$ by $2$ .

$\sqrt{{(- \frac{8 \cdot 2}{6} + \frac{1}{2} \cdot \frac{3}{3})}^{2} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.4.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$\sqrt{{(- \frac{8 \cdot 2}{6} + \frac{3}{2 \cdot 3})}^{2} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.4.4

Multiply $2$ by $3$ .

$\sqrt{{(- \frac{8 \cdot 2}{6} + \frac{3}{6})}^{2} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.5

Combine the numerators over the common denominator.

$\sqrt{{(\frac{- 8 \cdot 2 + 3}{6})}^{2} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 8$ by $2$ .

$\sqrt{{(\frac{- 16 + 3}{6})}^{2} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.6.2

Add $- 16$ and $3$ .

$\sqrt{{(\frac{- 13}{6})}^{2} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.7

Move the negative in front of the fraction.

$\sqrt{{(- \frac{13}{6})}^{2} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.8

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

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Step 3.8.1

Apply the product rule to $- \frac{13}{6}$ .

$\sqrt{{(- 1)}^{2} {(\frac{13}{6})}^{2} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.8.2

Apply the product rule to $\frac{13}{6}$ .

$\sqrt{{(- 1)}^{2} \frac{13^{2}}{6^{2}} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.9

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

$\sqrt{1 \frac{13^{2}}{6^{2}} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.10

Multiply $\frac{13^{2}}{6^{2}}$ by $1$ .

$\sqrt{\frac{13^{2}}{6^{2}} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.11

Raise $13$ to the power of $2$ .

$\sqrt{\frac{169}{6^{2}} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.12

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

$\sqrt{\frac{169}{36} + {(\frac{1}{21} - \frac{5}{7})}^{2}}$

Step 3.13

To write $- \frac{5}{7}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt{\frac{169}{36} + {(\frac{1}{21} - \frac{5}{7} \cdot \frac{3}{3})}^{2}}$

Step 3.14

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.14.1

Multiply $\frac{5}{7}$ by $\frac{3}{3}$ .

$\sqrt{\frac{169}{36} + {(\frac{1}{21} - \frac{5 \cdot 3}{7 \cdot 3})}^{2}}$

Step 3.14.2

Multiply $7$ by $3$ .

$\sqrt{\frac{169}{36} + {(\frac{1}{21} - \frac{5 \cdot 3}{21})}^{2}}$

Step 3.15

Combine the numerators over the common denominator.

$\sqrt{\frac{169}{36} + {(\frac{1 - 5 \cdot 3}{21})}^{2}}$

Step 3.16

Simplify the numerator.

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Step 3.16.1

Multiply $- 5$ by $3$ .

$\sqrt{\frac{169}{36} + {(\frac{1 - 15}{21})}^{2}}$

Step 3.16.2

Subtract $15$ from $1$ .

$\sqrt{\frac{169}{36} + {(\frac{- 14}{21})}^{2}}$

Step 3.17

Cancel the common factor of $- 14$ and $21$ .

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Step 3.17.1

Factor $7$ out of $- 14$ .

$\sqrt{\frac{169}{36} + {(\frac{7 (- 2)}{21})}^{2}}$

Step 3.17.2

Cancel the common factors.

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Step 3.17.2.1

Factor $7$ out of $21$ .

$\sqrt{\frac{169}{36} + {(\frac{7 \cdot - 2}{7 \cdot 3})}^{2}}$

Step 3.17.2.2

Cancel the common factor.

$\sqrt{\frac{169}{36} + {(\frac{7 \cdot - 2}{7 \cdot 3})}^{2}}$

Step 3.17.2.3

Rewrite the expression.

$\sqrt{\frac{169}{36} + {(\frac{- 2}{3})}^{2}}$

Step 3.18

Move the negative in front of the fraction.

$\sqrt{\frac{169}{36} + {(- \frac{2}{3})}^{2}}$

Step 3.19

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

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Step 3.19.1

Apply the product rule to $- \frac{2}{3}$ .

$\sqrt{\frac{169}{36} + {(- 1)}^{2} {(\frac{2}{3})}^{2}}$

Step 3.19.2

Apply the product rule to $\frac{2}{3}$ .

$\sqrt{\frac{169}{36} + {(- 1)}^{2} \frac{2^{2}}{3^{2}}}$

Step 3.20

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

$\sqrt{\frac{169}{36} + 1 \frac{2^{2}}{3^{2}}}$

Step 3.21

Multiply $\frac{2^{2}}{3^{2}}$ by $1$ .

$\sqrt{\frac{169}{36} + \frac{2^{2}}{3^{2}}}$

Step 3.22

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

$\sqrt{\frac{169}{36} + \frac{4}{3^{2}}}$

Step 3.23

Raise $3$ to the power of $2$ .

$\sqrt{\frac{169}{36} + \frac{4}{9}}$

Step 3.24

To write $\frac{4}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{169}{36} + \frac{4}{9} \cdot \frac{4}{4}}$

Step 3.25

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.25.1

Multiply $\frac{4}{9}$ by $\frac{4}{4}$ .

$\sqrt{\frac{169}{36} + \frac{4 \cdot 4}{9 \cdot 4}}$

Step 3.25.2

Multiply $9$ by $4$ .

$\sqrt{\frac{169}{36} + \frac{4 \cdot 4}{36}}$

Step 3.26

Combine the numerators over the common denominator.

$\sqrt{\frac{169 + 4 \cdot 4}{36}}$

Step 3.27

Simplify the numerator.

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Step 3.27.1

Multiply $4$ by $4$ .

$\sqrt{\frac{169 + 16}{36}}$

Step 3.27.2

Add $169$ and $16$ .

$\sqrt{\frac{185}{36}}$

Step 3.28

Rewrite $\sqrt{\frac{185}{36}}$ as $\frac{\sqrt{185}}{\sqrt{36}}$ .

$\frac{\sqrt{185}}{\sqrt{36}}$

Step 3.29

Simplify the denominator.

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Step 3.29.1

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

$\frac{\sqrt{185}}{\sqrt{6^{2}}}$

Step 3.29.2

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

$\frac{\sqrt{185}}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{185}}{6}$

Decimal Form:

$2.26691175 \dots$

Step 5