Basic Math Examples

$(- \frac{2}{3}, - \frac{1}{5})$ , $(\frac{5}{6}, - \frac{3}{10})$

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.

$\sqrt{{(\frac{5}{6} - (- \frac{2}{3}))}^{2} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3

Simplify.

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Step 3.1

Multiply $- (- \frac{2}{3})$ .

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Step 3.1.1

Multiply $- 1$ by $- 1$ .

$\sqrt{{(\frac{5}{6} + 1 (\frac{2}{3}))}^{2} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.1.2

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

$\sqrt{{(\frac{5}{6} + \frac{2}{3})}^{2} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.2

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

$\sqrt{{(\frac{5}{6} + \frac{2}{3} \cdot \frac{2}{2})}^{2} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.3

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

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Step 3.3.1

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

$\sqrt{{(\frac{5}{6} + \frac{2 \cdot 2}{3 \cdot 2})}^{2} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.3.2

Multiply $3$ by $2$ .

$\sqrt{{(\frac{5}{6} + \frac{2 \cdot 2}{6})}^{2} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.4

Combine the numerators over the common denominator.

$\sqrt{{(\frac{5 + 2 \cdot 2}{6})}^{2} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2$ by $2$ .

$\sqrt{{(\frac{5 + 4}{6})}^{2} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.5.2

Add $5$ and $4$ .

$\sqrt{{(\frac{9}{6})}^{2} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.6

Cancel the common factor of $9$ and $6$ .

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Step 3.6.1

Factor $3$ out of $9$ .

$\sqrt{{(\frac{3 (3)}{6})}^{2} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $3$ out of $6$ .

$\sqrt{{(\frac{3 \cdot 3}{3 \cdot 2})}^{2} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.6.2.2

Cancel the common factor.

$\sqrt{{(\frac{3 \cdot 3}{3 \cdot 2})}^{2} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.6.2.3

Rewrite the expression.

$\sqrt{{(\frac{3}{2})}^{2} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.7

Simplify the expression.

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Step 3.7.1

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

$\sqrt{\frac{3^{2}}{2^{2}} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.7.2

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

$\sqrt{\frac{9}{2^{2}} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.7.3

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

$\sqrt{\frac{9}{4} + {((- \frac{3}{10}) - (- \frac{1}{5}))}^{2}}$

Step 3.8

Multiply $- (- \frac{1}{5})$ .

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Step 3.8.1

Multiply $- 1$ by $- 1$ .

$\sqrt{\frac{9}{4} + {(- \frac{3}{10} + 1 (\frac{1}{5}))}^{2}}$

Step 3.8.2

Multiply $\frac{1}{5}$ by $1$ .

$\sqrt{\frac{9}{4} + {(- \frac{3}{10} + \frac{1}{5})}^{2}}$

Step 3.9

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{9}{4} + {(- \frac{3}{10} + \frac{1}{5} \cdot \frac{2}{2})}^{2}}$

Step 3.10

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.10.1

Multiply $\frac{1}{5}$ by $\frac{2}{2}$ .

$\sqrt{\frac{9}{4} + {(- \frac{3}{10} + \frac{2}{5 \cdot 2})}^{2}}$

Step 3.10.2

Multiply $5$ by $2$ .

$\sqrt{\frac{9}{4} + {(- \frac{3}{10} + \frac{2}{10})}^{2}}$

Step 3.11

Combine the numerators over the common denominator.

$\sqrt{\frac{9}{4} + {(\frac{- 3 + 2}{10})}^{2}}$

Step 3.12

Add $- 3$ and $2$ .

$\sqrt{\frac{9}{4} + {(\frac{- 1}{10})}^{2}}$

Step 3.13

Move the negative in front of the fraction.

$\sqrt{\frac{9}{4} + {(- \frac{1}{10})}^{2}}$

Step 3.14

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

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Step 3.14.1

Apply the product rule to $- \frac{1}{10}$ .

$\sqrt{\frac{9}{4} + {(- 1)}^{2} {(\frac{1}{10})}^{2}}$

Step 3.14.2

Apply the product rule to $\frac{1}{10}$ .

$\sqrt{\frac{9}{4} + {(- 1)}^{2} \frac{1^{2}}{10^{2}}}$

Step 3.15

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

$\sqrt{\frac{9}{4} + 1 \frac{1^{2}}{10^{2}}}$

Step 3.16

Multiply $\frac{1^{2}}{10^{2}}$ by $1$ .

$\sqrt{\frac{9}{4} + \frac{1^{2}}{10^{2}}}$

Step 3.17

One to any power is one.

$\sqrt{\frac{9}{4} + \frac{1}{10^{2}}}$

Step 3.18

Raise $10$ to the power of $2$ .

$\sqrt{\frac{9}{4} + \frac{1}{100}}$

Step 3.19

To write $\frac{9}{4}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\sqrt{\frac{9}{4} \cdot \frac{25}{25} + \frac{1}{100}}$

Step 3.20

Write each expression with a common denominator of $100$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.20.1

Multiply $\frac{9}{4}$ by $\frac{25}{25}$ .

$\sqrt{\frac{9 \cdot 25}{4 \cdot 25} + \frac{1}{100}}$

Step 3.20.2

Multiply $4$ by $25$ .

$\sqrt{\frac{9 \cdot 25}{100} + \frac{1}{100}}$

Step 3.21

Combine the numerators over the common denominator.

$\sqrt{\frac{9 \cdot 25 + 1}{100}}$

Step 3.22

Simplify the numerator.

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Step 3.22.1

Multiply $9$ by $25$ .

$\sqrt{\frac{225 + 1}{100}}$

Step 3.22.2

Add $225$ and $1$ .

$\sqrt{\frac{226}{100}}$

Step 3.23

Cancel the common factor of $226$ and $100$ .

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Step 3.23.1

Factor $2$ out of $226$ .

$\sqrt{\frac{2 (113)}{100}}$

Step 3.23.2

Cancel the common factors.

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Step 3.23.2.1

Factor $2$ out of $100$ .

$\sqrt{\frac{2 \cdot 113}{2 \cdot 50}}$

Step 3.23.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 113}{2 \cdot 50}}$

Step 3.23.2.3

Rewrite the expression.

$\sqrt{\frac{113}{50}}$

Step 3.24

Rewrite $\sqrt{\frac{113}{50}}$ as $\frac{\sqrt{113}}{\sqrt{50}}$ .

$\frac{\sqrt{113}}{\sqrt{50}}$

Step 3.25

Simplify the denominator.

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Step 3.25.1

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 3.25.1.1

Factor $25$ out of $50$ .

$\frac{\sqrt{113}}{\sqrt{25 (2)}}$

Step 3.25.1.2

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{113}}{\sqrt{5^{2} \cdot 2}}$

Step 3.25.2

Pull terms out from under the radical.

$\frac{\sqrt{113}}{5 \sqrt{2}}$

Step 3.26

Multiply $\frac{\sqrt{113}}{5 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{113}}{5 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3.27

Combine and simplify the denominator.

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Step 3.27.1

Multiply $\frac{\sqrt{113}}{5 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{113} \sqrt{2}}{5 \sqrt{2} \sqrt{2}}$

Step 3.27.2

Move $\sqrt{2}$ .

$\frac{\sqrt{113} \sqrt{2}}{5 (\sqrt{2} \sqrt{2})}$

Step 3.27.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{113} \sqrt{2}}{5 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 3.27.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{113} \sqrt{2}}{5 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 3.27.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{113} \sqrt{2}}{5 {\sqrt{2}}^{1 + 1}}$

Step 3.27.6

Add $1$ and $1$ .

$\frac{\sqrt{113} \sqrt{2}}{5 {\sqrt{2}}^{2}}$

Step 3.27.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.27.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\sqrt{113} \sqrt{2}}{5 {(2^{\frac{1}{2}})}^{2}}$

Step 3.27.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{113} \sqrt{2}}{5 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 3.27.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{113} \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}}$

Step 3.27.7.4

Cancel the common factor of $2$ .

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Step 3.27.7.4.1

Cancel the common factor.

$\frac{\sqrt{113} \sqrt{2}}{5 \cdot 2^{\frac{2}{2}}}$

Step 3.27.7.4.2

Rewrite the expression.

$\frac{\sqrt{113} \sqrt{2}}{5 \cdot 2^{1}}$

Step 3.27.7.5

Evaluate the exponent.

$\frac{\sqrt{113} \sqrt{2}}{5 \cdot 2}$

Step 3.28

Simplify the numerator.

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Step 3.28.1

Combine using the product rule for radicals.

$\frac{\sqrt{113 \cdot 2}}{5 \cdot 2}$

Step 3.28.2

Multiply $113$ by $2$ .

$\frac{\sqrt{226}}{5 \cdot 2}$

Step 3.29

Multiply $5$ by $2$ .

$\frac{\sqrt{226}}{10}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{226}}{10}$

Decimal Form:

$1.50332963 \dots$

Step 5