Algebra Examples

$(4, 1 \frac{1}{2})$ , $(2 \frac{2}{3}, - 3)$

Step 1

Convert $1 \frac{1}{2}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

$(4, 1 + \frac{1}{2}) - (2 \frac{2}{3}, - 3)$

Step 1.2

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

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

$(4, \frac{2}{2} + \frac{1}{2}) - (2 \frac{2}{3}, - 3)$

Step 1.2.2

Combine the numerators over the common denominator.

$(4, \frac{2 + 1}{2}) - (2 \frac{2}{3}, - 3)$

Step 1.2.3

Add $2$ and $1$ .

$(4, \frac{3}{2}) - (2 \frac{2}{3}, - 3)$

Step 2

Convert $2 \frac{2}{3}$ to an improper fraction.

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Step 2.1

A mixed number is an addition of its whole and fractional parts.

$(4, \frac{3}{2}) - (2 + \frac{2}{3}, - 3)$

Step 2.2

Add $2$ and $\frac{2}{3}$ .

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Step 2.2.1

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

$(4, \frac{3}{2}) - (2 \cdot \frac{3}{3} + \frac{2}{3}, - 3)$

Step 2.2.2

Combine $2$ and $\frac{3}{3}$ .

$(4, \frac{3}{2}) - (\frac{2 \cdot 3}{3} + \frac{2}{3}, - 3)$

Step 2.2.3

Combine the numerators over the common denominator.

$(4, \frac{3}{2}) - (\frac{2 \cdot 3 + 2}{3}, - 3)$

Step 2.2.4

Simplify the numerator.

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Step 2.2.4.1

Multiply $2$ by $3$ .

$(4, \frac{3}{2}) - (\frac{6 + 2}{3}, - 3)$

Step 2.2.4.2

Add $6$ and $2$ .

$(4, \frac{3}{2}) - (\frac{8}{3}, - 3)$

Step 3

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 4

Substitute the actual values of the points into the distance formula.

$\sqrt{{(\frac{8}{3} - 4)}^{2} + {((- 3) - \frac{3}{2})}^{2}}$

Step 5

Simplify.

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Step 5.1

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt{{(\frac{8}{3} - 4 \cdot \frac{3}{3})}^{2} + {((- 3) - \frac{3}{2})}^{2}}$

Step 5.2

Combine $- 4$ and $\frac{3}{3}$ .

$\sqrt{{(\frac{8}{3} + \frac{- 4 \cdot 3}{3})}^{2} + {((- 3) - \frac{3}{2})}^{2}}$

Step 5.3

Combine the numerators over the common denominator.

$\sqrt{{(\frac{8 - 4 \cdot 3}{3})}^{2} + {((- 3) - \frac{3}{2})}^{2}}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $- 4$ by $3$ .

$\sqrt{{(\frac{8 - 12}{3})}^{2} + {((- 3) - \frac{3}{2})}^{2}}$

Step 5.4.2

Subtract $12$ from $8$ .

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

Step 5.5

Move the negative in front of the fraction.

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

Step 5.6

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

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Step 5.6.1

Apply the product rule to $- \frac{4}{3}$ .

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

Step 5.6.2

Apply the product rule to $\frac{4}{3}$ .

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

Step 5.7

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

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

Step 5.8

Multiply $\frac{4^{2}}{3^{2}}$ by $1$ .

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

Step 5.9

Raise $4$ to the power of $2$ .

$\sqrt{\frac{16}{3^{2}} + {((- 3) - \frac{3}{2})}^{2}}$

Step 5.10

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

$\sqrt{\frac{16}{9} + {((- 3) - \frac{3}{2})}^{2}}$

Step 5.11

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

$\sqrt{\frac{16}{9} + {(- 3 \cdot \frac{2}{2} - \frac{3}{2})}^{2}}$

Step 5.12

Combine $- 3$ and $\frac{2}{2}$ .

$\sqrt{\frac{16}{9} + {(\frac{- 3 \cdot 2}{2} - \frac{3}{2})}^{2}}$

Step 5.13

Combine the numerators over the common denominator.

$\sqrt{\frac{16}{9} + {(\frac{- 3 \cdot 2 - 3}{2})}^{2}}$

Step 5.14

Simplify the numerator.

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Step 5.14.1

Multiply $- 3$ by $2$ .

$\sqrt{\frac{16}{9} + {(\frac{- 6 - 3}{2})}^{2}}$

Step 5.14.2

Subtract $3$ from $- 6$ .

$\sqrt{\frac{16}{9} + {(\frac{- 9}{2})}^{2}}$

Step 5.15

Move the negative in front of the fraction.

$\sqrt{\frac{16}{9} + {(- \frac{9}{2})}^{2}}$

Step 5.16

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

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Step 5.16.1

Apply the product rule to $- \frac{9}{2}$ .

$\sqrt{\frac{16}{9} + {(- 1)}^{2} {(\frac{9}{2})}^{2}}$

Step 5.16.2

Apply the product rule to $\frac{9}{2}$ .

$\sqrt{\frac{16}{9} + {(- 1)}^{2} \frac{9^{2}}{2^{2}}}$

Step 5.17

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

$\sqrt{\frac{16}{9} + 1 \frac{9^{2}}{2^{2}}}$

Step 5.18

Multiply $\frac{9^{2}}{2^{2}}$ by $1$ .

$\sqrt{\frac{16}{9} + \frac{9^{2}}{2^{2}}}$

Step 5.19

Raise $9$ to the power of $2$ .

$\sqrt{\frac{16}{9} + \frac{81}{2^{2}}}$

Step 5.20

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

$\sqrt{\frac{16}{9} + \frac{81}{4}}$

Step 5.21

To write $\frac{16}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\sqrt{\frac{16}{9} \cdot \frac{4}{4} + \frac{81}{4}}$

Step 5.22

To write $\frac{81}{4}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\sqrt{\frac{16}{9} \cdot \frac{4}{4} + \frac{81}{4} \cdot \frac{9}{9}}$

Step 5.23

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

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Step 5.23.1

Multiply $\frac{16}{9}$ by $\frac{4}{4}$ .

$\sqrt{\frac{16 \cdot 4}{9 \cdot 4} + \frac{81}{4} \cdot \frac{9}{9}}$

Step 5.23.2

Multiply $9$ by $4$ .

$\sqrt{\frac{16 \cdot 4}{36} + \frac{81}{4} \cdot \frac{9}{9}}$

Step 5.23.3

Multiply $\frac{81}{4}$ by $\frac{9}{9}$ .

$\sqrt{\frac{16 \cdot 4}{36} + \frac{81 \cdot 9}{4 \cdot 9}}$

Step 5.23.4

Multiply $4$ by $9$ .

$\sqrt{\frac{16 \cdot 4}{36} + \frac{81 \cdot 9}{36}}$

Step 5.24

Combine the numerators over the common denominator.

$\sqrt{\frac{16 \cdot 4 + 81 \cdot 9}{36}}$

Step 5.25

Simplify the numerator.

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Step 5.25.1

Multiply $16$ by $4$ .

$\sqrt{\frac{64 + 81 \cdot 9}{36}}$

Step 5.25.2

Multiply $81$ by $9$ .

$\sqrt{\frac{64 + 729}{36}}$

Step 5.25.3

Add $64$ and $729$ .

$\sqrt{\frac{793}{36}}$

Step 5.26

Rewrite $\sqrt{\frac{793}{36}}$ as $\frac{\sqrt{793}}{\sqrt{36}}$ .

$\frac{\sqrt{793}}{\sqrt{36}}$

Step 5.27

Simplify the denominator.

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Step 5.27.1

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

$\frac{\sqrt{793}}{\sqrt{6^{2}}}$

Step 5.27.2

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

$\frac{\sqrt{793}}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{793}}{6}$

Decimal Form:

$4.69337594 \dots$

Step 7