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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