Find the Distance Between Two Points (0,- square root of 14) , ( square root of 31,0)

$(0, - \sqrt{14})$ , $(\sqrt{31}, 0)$

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.

$\sqrt{{(\sqrt{31} - 0)}^{2} + {(0 - (- \sqrt{14}))}^{2}}$

Step 3

Simplify.

Subtract $0$ from $\sqrt{31}$ .

$\sqrt{{\sqrt{31}}^{2} + {(0 - (- \sqrt{14}))}^{2}}$

Rewrite ${\sqrt{31}}^{2}$ as $31$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{31}$ as $31^{\frac{1}{2}}$ .

$\sqrt{{(31^{\frac{1}{2}})}^{2} + {(0 - (- \sqrt{14}))}^{2}}$

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

$\sqrt{31^{\frac{1}{2} \cdot 2} + {(0 - (- \sqrt{14}))}^{2}}$

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

$\sqrt{31^{\frac{2}{2}} + {(0 - (- \sqrt{14}))}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{31^{\frac{2}{2}} + {(0 - (- \sqrt{14}))}^{2}}$

Rewrite the expression.

$\sqrt{31^{1} + {(0 - (- \sqrt{14}))}^{2}}$

Evaluate the exponent.

$\sqrt{31 + {(0 - (- \sqrt{14}))}^{2}}$

Multiply $- (- \sqrt{14})$ .

Multiply $- 1$ by $- 1$ .

$\sqrt{31 + {(0 + 1 \sqrt{14})}^{2}}$

Multiply $\sqrt{14}$ by $1$ .

$\sqrt{31 + {(0 + \sqrt{14})}^{2}}$

Add $0$ and $\sqrt{14}$ .

$\sqrt{31 + {\sqrt{14}}^{2}}$

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

$\sqrt{31 + {(14^{\frac{1}{2}})}^{2}}$

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

$\sqrt{31 + 14^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{31 + 14^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{31 + 14^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{31 + 14^{1}}$

Evaluate the exponent.

$\sqrt{31 + 14}$

Add $31$ and $14$ .

$\sqrt{45}$

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$\sqrt{9 (5)}$

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2} \cdot 5}$

Pull terms out from under the radical.

$3 \sqrt{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{5}$

Decimal Form:

$6.70820393 \dots$

Step 5

Find the Distance Between Two Points (0,- square root of 14) , ( square root of 31,0)

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