Find the Distance Between Two Points ( square root of 2,0) , (0,- square root of 2)

$(\sqrt{2}, 0)$ , $(0, - \sqrt{2})$

Step 1

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 2

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

$\sqrt{{(0 - \sqrt{2})}^{2} + {((- \sqrt{2}) - 0)}^{2}}$

Step 3

Simplify.

Simplify the expression.

Subtract $\sqrt{2}$ from $0$ .

$\sqrt{{(- \sqrt{2})}^{2} + {((- \sqrt{2}) - 0)}^{2}}$

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

$\sqrt{{(- 1)}^{2} {\sqrt{2}}^{2} + {((- \sqrt{2}) - 0)}^{2}}$

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

$\sqrt{1 {\sqrt{2}}^{2} + {((- \sqrt{2}) - 0)}^{2}}$

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$\sqrt{{\sqrt{2}}^{2} + {((- \sqrt{2}) - 0)}^{2}}$

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sqrt{{(2^{\frac{1}{2}})}^{2} + {((- \sqrt{2}) - 0)}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{2^{\frac{1}{2} \cdot 2} + {((- \sqrt{2}) - 0)}^{2}}$

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

$\sqrt{2^{\frac{2}{2}} + {((- \sqrt{2}) - 0)}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{2^{\frac{2}{2}} + {((- \sqrt{2}) - 0)}^{2}}$

Rewrite the expression.

$\sqrt{2^{1} + {((- \sqrt{2}) - 0)}^{2}}$

Evaluate the exponent.

$\sqrt{2 + {((- \sqrt{2}) - 0)}^{2}}$

Simplify the expression.

Subtract $0$ from $- \sqrt{2}$ .

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

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

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

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

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

Multiply ${\sqrt{2}}^{2}$ by $1$ .

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$\sqrt{2 + 2}$

Simplify the expression.

Add $2$ and $2$ .

$\sqrt{4}$

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

$\sqrt{2^{2}}$

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

$2$

Step 4

Find the Distance Between Two Points ( square root of 2,0) , (0,- square root of 2)

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