Find the Distance Between Two Points (-4 square root of 2,- square root of 8) , (-5 square root of 2, square root of 18)

$(- 4 \sqrt{2}, - \sqrt{8})$ ,

$(- 5 \sqrt{2}, \sqrt{18})$

Step 1

Rewrite

$8$ as

$2^{2} \cdot 2$ .

Factor

$4$ out of

$8$ .

$(- 4 \sqrt{2}, - \sqrt{4 (2)}) - (- 5 \sqrt{2}, \sqrt{18})$

Rewrite

$4$ as

$2^{2}$ .

$(- 4 \sqrt{2}, - \sqrt{2^{2} \cdot 2}) - (- 5 \sqrt{2}, \sqrt{18})$

Step 2

Pull terms out from under the radical.

$(- 4 \sqrt{2}, - (2 \sqrt{2})) - (- 5 \sqrt{2}, \sqrt{18})$

Step 3

Multiply

$2$ by

$- 1$ .

$(- 4 \sqrt{2}, - 2 \sqrt{2}) - (- 5 \sqrt{2}, \sqrt{18})$

Step 4

Rewrite

$18$ as

$3^{2} \cdot 2$ .

Factor

$9$ out of

$18$ .

$(- 4 \sqrt{2}, - 2 \sqrt{2}) - (- 5 \sqrt{2}, \sqrt{9 (2)})$

Rewrite

$9$ as

$3^{2}$ .

$(- 4 \sqrt{2}, - 2 \sqrt{2}) - (- 5 \sqrt{2}, \sqrt{3^{2} \cdot 2})$

Step 5

Pull terms out from under the radical.

$(- 4 \sqrt{2}, - 2 \sqrt{2}) - (- 5 \sqrt{2}, 3 \sqrt{2})$

Step 6

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 7

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

$\sqrt{{((- 5 \sqrt{2}) - (- 4 \sqrt{2}))}^{2} + {(3 \sqrt{2} - (- 2 \sqrt{2}))}^{2}}$

Step 8

Simplify.

Multiply

$- 4$ by

$- 1$ .

$\sqrt{{(- 5 \sqrt{2} + 4 \sqrt{2})}^{2} + {(3 \sqrt{2} - (- 2 \sqrt{2}))}^{2}}$

Add

$- 5 \sqrt{2}$ and

$4 \sqrt{2}$ .

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

Simplify the expression.

Apply the product rule to

$- \sqrt{2}$ .

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

Raise

$- 1$ to the power of

$2$ .

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

Multiply

${\sqrt{2}}^{2}$ by

$1$ .

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

Rewrite

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

$2$ .

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Combine

$\frac{1}{2}$ and

$2$ .

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

Cancel the common factor of

$2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply

$- 2$ by

$- 1$ .

$\sqrt{2 + {(3 \sqrt{2} + 2 \sqrt{2})}^{2}}$

Add

$3 \sqrt{2}$ and

$2 \sqrt{2}$ .

$\sqrt{2 + {(5 \sqrt{2})}^{2}}$

Simplify the expression.

Apply the product rule to

$5 \sqrt{2}$ .

$\sqrt{2 + 5^{2} {\sqrt{2}}^{2}}$

Raise

$5$ to the power of

$2$ .

$\sqrt{2 + 25 {\sqrt{2}}^{2}}$

Rewrite

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

$2$ .

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$\sqrt{2 + 25 {(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{2 + 25 \cdot 2^{\frac{1}{2} \cdot 2}}$

Combine

$\frac{1}{2}$ and

$2$ .

$\sqrt{2 + 25 \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of

$2$ .

Cancel the common factor.

$\sqrt{2 + 25 \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{2 + 25 \cdot 2^{1}}$

Evaluate the exponent.

$\sqrt{2 + 25 \cdot 2}$

Simplify the expression.

Multiply

$25$ by

$2$ .

$\sqrt{2 + 50}$

Add

$2$ and

$50$ .

$\sqrt{52}$

Rewrite

$52$ as

$2^{2} \cdot 13$ .

Factor

$4$ out of

$52$ .

$\sqrt{4 (13)}$

Rewrite

$4$ as

$2^{2}$ .

$\sqrt{2^{2} \cdot 13}$

Pull terms out from under the radical.

$2 \sqrt{13}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{13}$

Decimal Form:

$7.21110255 \dots$

Step 10

Find the Distance Between Two Points (-4 square root of 2,- square root of 8) , (-5 square root of 2, square root of 18)

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