Algebra Examples

$(12, 67, - 9)$ , $(\frac{1}{2}, 6, - 4)$

Step 1

To find the distance between two 3d points, square the difference of the x, y, and z points. Then, sum them and take the square root.

$\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2}}$

Step 2

Replace $x_{1}$ , $x_{2}$ , $y_{1}$ , $y_{2}$ , $z_{1}$ , and $z_{2}$ with the corresponding values.

$D i s t a n c e = \sqrt{{(\frac{1}{2} - 12)}^{2} + {(6 - 67)}^{2} + {(- 4 - (- 9))}^{2}}$

Step 3

Simplify the expression $\sqrt{{(\frac{1}{2} - 12)}^{2} + {(6 - 67)}^{2} + {(- 4 - (- 9))}^{2}}$ .

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Step 3.1

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$D i s t a n c e = \sqrt{{(\frac{1}{2} - 12 \cdot \frac{2}{2})}^{2} + {(6 - 67)}^{2} + {(- 4 - (- 9))}^{2}}$

Step 3.2

Combine $- 12$ and $\frac{2}{2}$ .

$D i s t a n c e = \sqrt{{(\frac{1}{2} + \frac{- 12 \cdot 2}{2})}^{2} + {(6 - 67)}^{2} + {(- 4 - (- 9))}^{2}}$

Step 3.3

Combine the numerators over the common denominator.

$D i s t a n c e = \sqrt{{(\frac{1 - 12 \cdot 2}{2})}^{2} + {(6 - 67)}^{2} + {(- 4 - (- 9))}^{2}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Multiply $- 12$ by $2$ .

$D i s t a n c e = \sqrt{{(\frac{1 - 24}{2})}^{2} + {(6 - 67)}^{2} + {(- 4 - (- 9))}^{2}}$

Step 3.4.2

Subtract $24$ from $1$ .

$D i s t a n c e = \sqrt{{(\frac{- 23}{2})}^{2} + {(6 - 67)}^{2} + {(- 4 - (- 9))}^{2}}$

Step 3.5

Move the negative in front of the fraction.

$D i s t a n c e = \sqrt{{(- \frac{23}{2})}^{2} + {(6 - 67)}^{2} + {(- 4 - (- 9))}^{2}}$

Step 3.6

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

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Step 3.6.1

Apply the product rule to $- \frac{23}{2}$ .

$D i s t a n c e = \sqrt{{(- 1)}^{2} {(\frac{23}{2})}^{2} + {(6 - 67)}^{2} + {(- 4 - (- 9))}^{2}}$

Step 3.6.2

Apply the product rule to $\frac{23}{2}$ .

$D i s t a n c e = \sqrt{{(- 1)}^{2} (\frac{23^{2}}{2^{2}}) + {(6 - 67)}^{2} + {(- 4 - (- 9))}^{2}}$

Step 3.7

Simplify the expression.

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Step 3.7.1

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

$D i s t a n c e = \sqrt{1 (\frac{23^{2}}{2^{2}}) + {(6 - 67)}^{2} + {(- 4 - (- 9))}^{2}}$

Step 3.7.2

Multiply $\frac{23^{2}}{2^{2}}$ by $1$ .

$D i s t a n c e = \sqrt{\frac{23^{2}}{2^{2}} + {(6 - 67)}^{2} + {(- 4 - (- 9))}^{2}}$

Step 3.7.3

Raise $23$ to the power of $2$ .

$D i s t a n c e = \sqrt{\frac{529}{2^{2}} + {(6 - 67)}^{2} + {(- 4 - (- 9))}^{2}}$

Step 3.7.4

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

$D i s t a n c e = \sqrt{\frac{529}{4} + {(6 - 67)}^{2} + {(- 4 - (- 9))}^{2}}$

Step 3.7.5

Subtract $67$ from $6$ .

$D i s t a n c e = \sqrt{\frac{529}{4} + {(- 61)}^{2} + {(- 4 - (- 9))}^{2}}$

Step 3.7.6

Raise $- 61$ to the power of $2$ .

$D i s t a n c e = \sqrt{\frac{529}{4} + 3721 + {(- 4 - (- 9))}^{2}}$

Step 3.8

Simplify each term.

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Step 3.8.1

Multiply $- 1$ by each element of the matrix.

$D i s t a n c e = \sqrt{\frac{529}{4} + 3721 + {(- 4 + 9)}^{2}}$

Step 3.8.2

Multiply $- 1$ by $- 9$ .

$D i s t a n c e = \sqrt{\frac{529}{4} + 3721 + {(- 4 + 9)}^{2}}$

Step 3.9

Simplify the expression.

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Step 3.9.1

Add $- 4$ and $9$ .

$D i s t a n c e = \sqrt{\frac{529}{4} + 3721 + 5^{2}}$

Step 3.9.2

Raise $5$ to the power of $2$ .

$D i s t a n c e = \sqrt{\frac{529}{4} + 3721 + 25}$

Step 3.10

To write $3721$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$D i s t a n c e = \sqrt{\frac{529}{4} + 3721 \cdot \frac{4}{4} + 25}$

Step 3.11

Combine $3721$ and $\frac{4}{4}$ .

$D i s t a n c e = \sqrt{\frac{529}{4} + \frac{3721 \cdot 4}{4} + 25}$

Step 3.12

Combine the numerators over the common denominator.

$D i s t a n c e = \sqrt{\frac{529 + 3721 \cdot 4}{4} + 25}$

Step 3.13

Simplify the numerator.

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Step 3.13.1

Multiply $3721$ by $4$ .

$D i s t a n c e = \sqrt{\frac{529 + 14884}{4} + 25}$

Step 3.13.2

Add $529$ and $14884$ .

$D i s t a n c e = \sqrt{\frac{15413}{4} + 25}$

Step 3.14

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

$D i s t a n c e = \sqrt{\frac{15413}{4} + 25 \cdot \frac{4}{4}}$

Step 3.15

Combine $25$ and $\frac{4}{4}$ .

$D i s t a n c e = \sqrt{\frac{15413}{4} + \frac{25 \cdot 4}{4}}$

Step 3.16

Combine the numerators over the common denominator.

$D i s t a n c e = \sqrt{\frac{15413 + 25 \cdot 4}{4}}$

Step 3.17

Simplify the numerator.

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Step 3.17.1

Multiply $25$ by $4$ .

$D i s t a n c e = \sqrt{\frac{15413 + 100}{4}}$

Step 3.17.2

Add $15413$ and $100$ .

$D i s t a n c e = \sqrt{\frac{15513}{4}}$

Step 3.18

Rewrite $\sqrt{\frac{15513}{4}}$ as $\frac{\sqrt{15513}}{\sqrt{4}}$ .

$D i s t a n c e = \frac{\sqrt{15513}}{\sqrt{4}}$

Step 3.19

Simplify the denominator.

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Step 3.19.1

Rewrite $4$ as $2^{2}$ .

$D i s t a n c e = \frac{\sqrt{15513}}{\sqrt{2^{2}}}$

Step 3.19.2

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

$D i s t a n c e = \frac{\sqrt{15513}}{2}$

Step 4

The distance between $(12, 67, - 9)$ and $(\frac{1}{2}, 6, - 4)$ is $\frac{\sqrt{15513}}{2}$ .

$\frac{\sqrt{15513}}{2} \approx 62.27559714$