Find the Distance (4/13,8,4/9) , (6/14,7/9,9)

$(\frac{4}{13}, 8, \frac{4}{9})$ , $(\frac{6}{14}, \frac{7}{9}, 9)$

Step 1

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

$D i s t a n c e = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2}}$

Step 2

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

$D i s t a n c e = \sqrt{{((\frac{6}{14}) - (\frac{4}{13}))}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Step 3

Cancel the common factor of $6$ and $14$ .

Factor $2$ out of $6$ .

$D i s t a n c e = \sqrt{{(\frac{2 (3)}{14} - (\frac{4}{13}))}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Cancel the common factors.

Factor $2$ out of $14$ .

$D i s t a n c e = \sqrt{{(\frac{2 \cdot 3}{2 \cdot 7} - (\frac{4}{13}))}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Cancel the common factor.

$D i s t a n c e = \sqrt{{(\frac{2 \cdot 3}{2 \cdot 7} - (\frac{4}{13}))}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Rewrite the expression.

$D i s t a n c e = \sqrt{{(\frac{3}{7} - (\frac{4}{13}))}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

$D i s t a n c e = \sqrt{{(\frac{3}{7} - \frac{4}{13})}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

To write $\frac{3}{7}$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$D i s t a n c e = \sqrt{{(\frac{3}{7} \cdot \frac{13}{13} - \frac{4}{13})}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

To write $- \frac{4}{13}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$D i s t a n c e = \sqrt{{(\frac{3}{7} \cdot \frac{13}{13} - \frac{4}{13} \cdot \frac{7}{7})}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Write each expression with a common denominator of $91$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{7}$ by $\frac{13}{13}$ .

$D i s t a n c e = \sqrt{{(\frac{3 \cdot 13}{7 \cdot 13} - \frac{4}{13} \cdot \frac{7}{7})}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Multiply $7$ by $13$ .

$D i s t a n c e = \sqrt{{(\frac{3 \cdot 13}{91} - \frac{4}{13} \cdot \frac{7}{7})}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Multiply $\frac{4}{13}$ by $\frac{7}{7}$ .

$D i s t a n c e = \sqrt{{(\frac{3 \cdot 13}{91} - \frac{4 \cdot 7}{13 \cdot 7})}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Multiply $13$ by $7$ .

$D i s t a n c e = \sqrt{{(\frac{3 \cdot 13}{91} - \frac{4 \cdot 7}{91})}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Combine the numerators over the common denominator.

$D i s t a n c e = \sqrt{{(\frac{3 \cdot 13 - 4 \cdot 7}{91})}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Simplify the numerator.

Multiply $3$ by $13$ .

$D i s t a n c e = \sqrt{{(\frac{39 - 4 \cdot 7}{91})}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Multiply $- 4$ by $7$ .

$D i s t a n c e = \sqrt{{(\frac{39 - 28}{91})}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Subtract $28$ from $39$ .

$D i s t a n c e = \sqrt{{(\frac{11}{91})}^{2} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Apply the product rule to $\frac{11}{91}$ .

$D i s t a n c e = \sqrt{\frac{11^{2}}{91^{2}} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Raise $11$ to the power of $2$ .

$D i s t a n c e = \sqrt{\frac{121}{91^{2}} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Raise $91$ to the power of $2$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + {((\frac{7}{9}) - (8))}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Multiply $- 1$ by $8$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + {(\frac{7}{9} - 8)}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + {(\frac{7}{9} - 8 \cdot \frac{9}{9})}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Combine $- 8$ and $\frac{9}{9}$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + {(\frac{7}{9} + \frac{- 8 \cdot 9}{9})}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Combine the numerators over the common denominator.

$D i s t a n c e = \sqrt{\frac{121}{8281} + {(\frac{7 - 8 \cdot 9}{9})}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Simplify the numerator.

Multiply $- 8$ by $9$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + {(\frac{7 - 72}{9})}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Subtract $72$ from $7$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + {(\frac{- 65}{9})}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Move the negative in front of the fraction.

$D i s t a n c e = \sqrt{\frac{121}{8281} + {(- \frac{65}{9})}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

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

Apply the product rule to $- \frac{65}{9}$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + {(- 1)}^{2} {(\frac{65}{9})}^{2} + {((9) - (\frac{4}{9}))}^{2}}$

Apply the product rule to $\frac{65}{9}$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + {(- 1)}^{2} (\frac{65^{2}}{9^{2}}) + {((9) - (\frac{4}{9}))}^{2}}$

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

$D i s t a n c e = \sqrt{\frac{121}{8281} + 1 (\frac{65^{2}}{9^{2}}) + {((9) - (\frac{4}{9}))}^{2}}$

Multiply $\frac{65^{2}}{9^{2}}$ by $1$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + \frac{65^{2}}{9^{2}} + {((9) - (\frac{4}{9}))}^{2}}$

Raise $65$ to the power of $2$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + \frac{4225}{9^{2}} + {((9) - (\frac{4}{9}))}^{2}}$

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

$D i s t a n c e = \sqrt{\frac{121}{8281} + \frac{4225}{81} + {((9) - (\frac{4}{9}))}^{2}}$

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

$D i s t a n c e = \sqrt{\frac{121}{8281} + \frac{4225}{81} + {(9 \cdot \frac{9}{9} - \frac{4}{9})}^{2}}$

Combine $9$ and $\frac{9}{9}$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + \frac{4225}{81} + {(\frac{9 \cdot 9}{9} - \frac{4}{9})}^{2}}$

Combine the numerators over the common denominator.

$D i s t a n c e = \sqrt{\frac{121}{8281} + \frac{4225}{81} + {(\frac{9 \cdot 9 - 4}{9})}^{2}}$

Simplify the numerator.

Multiply $9$ by $9$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + \frac{4225}{81} + {(\frac{81 - 4}{9})}^{2}}$

Subtract $4$ from $81$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + \frac{4225}{81} + {(\frac{77}{9})}^{2}}$

Apply the product rule to $\frac{77}{9}$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + \frac{4225}{81} + \frac{77^{2}}{9^{2}}}$

Raise $77$ to the power of $2$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} + \frac{4225}{81} + \frac{5929}{9^{2}}}$

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

$D i s t a n c e = \sqrt{\frac{121}{8281} + \frac{4225}{81} + \frac{5929}{81}}$

To write $\frac{121}{8281}$ as a fraction with a common denominator, multiply by $\frac{81}{81}$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} \cdot \frac{81}{81} + \frac{4225}{81} + \frac{5929}{81}}$

To write $\frac{4225}{81}$ as a fraction with a common denominator, multiply by $\frac{8281}{8281}$ .

$D i s t a n c e = \sqrt{\frac{121}{8281} \cdot \frac{81}{81} + \frac{4225}{81} \cdot \frac{8281}{8281} + \frac{5929}{81}}$

Write each expression with a common denominator of $670761$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{121}{8281}$ by $\frac{81}{81}$ .

$D i s t a n c e = \sqrt{\frac{121 \cdot 81}{8281 \cdot 81} + \frac{4225}{81} \cdot \frac{8281}{8281} + \frac{5929}{81}}$

Multiply $8281$ by $81$ .

$D i s t a n c e = \sqrt{\frac{121 \cdot 81}{670761} + \frac{4225}{81} \cdot \frac{8281}{8281} + \frac{5929}{81}}$

Multiply $\frac{4225}{81}$ by $\frac{8281}{8281}$ .

$D i s t a n c e = \sqrt{\frac{121 \cdot 81}{670761} + \frac{4225 \cdot 8281}{81 \cdot 8281} + \frac{5929}{81}}$

Multiply $81$ by $8281$ .

$D i s t a n c e = \sqrt{\frac{121 \cdot 81}{670761} + \frac{4225 \cdot 8281}{670761} + \frac{5929}{81}}$

Combine the numerators over the common denominator.

$D i s t a n c e = \sqrt{\frac{121 \cdot 81 + 4225 \cdot 8281}{670761} + \frac{5929}{81}}$

Simplify the numerator.

Multiply $121$ by $81$ .

$D i s t a n c e = \sqrt{\frac{9801 + 4225 \cdot 8281}{670761} + \frac{5929}{81}}$

Multiply $4225$ by $8281$ .

$D i s t a n c e = \sqrt{\frac{9801 + 34987225}{670761} + \frac{5929}{81}}$

Add $9801$ and $34987225$ .

$D i s t a n c e = \sqrt{\frac{34997026}{670761} + \frac{5929}{81}}$

To write $\frac{5929}{81}$ as a fraction with a common denominator, multiply by $\frac{8281}{8281}$ .

$D i s t a n c e = \sqrt{\frac{34997026}{670761} + \frac{5929}{81} \cdot \frac{8281}{8281}}$

Write each expression with a common denominator of $670761$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5929}{81}$ by $\frac{8281}{8281}$ .

$D i s t a n c e = \sqrt{\frac{34997026}{670761} + \frac{5929 \cdot 8281}{81 \cdot 8281}}$

Multiply $81$ by $8281$ .

$D i s t a n c e = \sqrt{\frac{34997026}{670761} + \frac{5929 \cdot 8281}{670761}}$

Combine the numerators over the common denominator.

$D i s t a n c e = \sqrt{\frac{34997026 + 5929 \cdot 8281}{670761}}$

Simplify the numerator.

Multiply $5929$ by $8281$ .

$D i s t a n c e = \sqrt{\frac{34997026 + 49098049}{670761}}$

Add $34997026$ and $49098049$ .

$D i s t a n c e = \sqrt{\frac{84095075}{670761}}$

Rewrite $\sqrt{\frac{84095075}{670761}}$ as $\frac{\sqrt{84095075}}{\sqrt{670761}}$ .

$D i s t a n c e = \frac{\sqrt{84095075}}{\sqrt{670761}}$

Simplify the numerator.

Rewrite $84095075$ as $5^{2} \cdot 3363803$ .

Factor $25$ out of $84095075$ .

$D i s t a n c e = \frac{\sqrt{25 (3363803)}}{\sqrt{670761}}$

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

$D i s t a n c e = \frac{\sqrt{5^{2} \cdot 3363803}}{\sqrt{670761}}$

Pull terms out from under the radical.

$D i s t a n c e = \frac{5 \sqrt{3363803}}{\sqrt{670761}}$

Simplify the denominator.

Rewrite $670761$ as $819^{2}$ .

$D i s t a n c e = \frac{5 \sqrt{3363803}}{\sqrt{819^{2}}}$

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

$D i s t a n c e = \frac{5 \sqrt{3363803}}{819}$

Step 4

The distance between $(\frac{4}{13}, 8, \frac{4}{9})$ and $(\frac{6}{14}, \frac{7}{9}, 9)$ is $\frac{5 \sqrt{3363803}}{819}$ .

$\frac{5 \sqrt{3363803}}{819} \approx 11.19699229$

Find the Distance (4/13,8,4/9) , (6/14,7/9,9)

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