Find the Distance Between Two Points ( square root of 3,- square root of 7) and (-7 square root of 3,-5 square root of 7)

$(\sqrt{3}, - \sqrt{7})$ and $(- 7 \sqrt{3}, - 5 \sqrt{7})$

Use the distance formula to determine the distance between the two points.

$Distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

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

$\sqrt{{((- 7 \sqrt{3}) - \sqrt{3})}^{2} + {((- 5 \sqrt{7}) - (- \sqrt{7}))}^{2}}$

Simplify.

Subtract $\sqrt{3}$ from $- 7 \sqrt{3}$ .

$\sqrt{{(- 8 \sqrt{3})}^{2} + {((- 5 \sqrt{7}) - (- \sqrt{7}))}^{2}}$

Simplify the expression.

Apply the product rule to $- 8 \sqrt{3}$ .

$\sqrt{{(- 8)}^{2} {\sqrt{3}}^{2} + {((- 5 \sqrt{7}) - (- \sqrt{7}))}^{2}}$

Raise $- 8$ to the power of $2$ .

$\sqrt{64 {\sqrt{3}}^{2} + {((- 5 \sqrt{7}) - (- \sqrt{7}))}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{64 {(3^{\frac{1}{2}})}^{2} + {((- 5 \sqrt{7}) - (- \sqrt{7}))}^{2}}$

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

$\sqrt{64 \cdot 3^{\frac{1}{2} \cdot 2} + {((- 5 \sqrt{7}) - (- \sqrt{7}))}^{2}}$

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

$\sqrt{64 \cdot 3^{\frac{2}{2}} + {((- 5 \sqrt{7}) - (- \sqrt{7}))}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{64 \cdot 3^{\frac{2}{2}} + {((- 5 \sqrt{7}) - (- \sqrt{7}))}^{2}}$

Rewrite the expression.

$\sqrt{64 \cdot 3^{1} + {((- 5 \sqrt{7}) - (- \sqrt{7}))}^{2}}$

Evaluate the exponent.

$\sqrt{64 \cdot 3 + {((- 5 \sqrt{7}) - (- \sqrt{7}))}^{2}}$

Multiply $64$ by $3$ .

$\sqrt{192 + {((- 5 \sqrt{7}) - (- \sqrt{7}))}^{2}}$

Multiply $- (- \sqrt{7})$ .

Multiply $- 1$ by $- 1$ .

$\sqrt{192 + {(- 5 \sqrt{7} + 1 \sqrt{7})}^{2}}$

Multiply $\sqrt{7}$ by $1$ .

$\sqrt{192 + {(- 5 \sqrt{7} + \sqrt{7})}^{2}}$

Add $- 5 \sqrt{7}$ and $\sqrt{7}$ .

$\sqrt{192 + {(- 4 \sqrt{7})}^{2}}$

Simplify the expression.

Apply the product rule to $- 4 \sqrt{7}$ .

$\sqrt{192 + {(- 4)}^{2} {\sqrt{7}}^{2}}$

Raise $- 4$ to the power of $2$ .

$\sqrt{192 + 16 {\sqrt{7}}^{2}}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{192 + 16 {(7^{\frac{1}{2}})}^{2}}$

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

$\sqrt{192 + 16 \cdot 7^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{192 + 16 \cdot 7^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{192 + 16 \cdot 7^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{192 + 16 \cdot 7^{1}}$

Evaluate the exponent.

$\sqrt{192 + 16 \cdot 7}$

Simplify the expression.

Multiply $16$ by $7$ .

$\sqrt{192 + 112}$

Add $192$ and $112$ .

$\sqrt{304}$

Rewrite $304$ as $4^{2} \cdot 19$ .

Factor $16$ out of $304$ .

$\sqrt{16 (19)}$

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2} \cdot 19}$

Pull terms out from under the radical.

$4 \sqrt{19}$

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{19}$

Decimal Form:

$17.43559577 \dots$

Find the Distance Between Two Points ( square root of 3,- square root of 7) and (-7 square root of 3,-5 square root of 7)

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