Find the Distance Between Two Points ( square root of 5,- square root of 3) , (4 square root of 5,-7 square root of 3)

$(\sqrt{5}, - \sqrt{3})$ , $(4 \sqrt{5}, - 7 \sqrt{3})$

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.

$\sqrt{{(4 \sqrt{5} - \sqrt{5})}^{2} + {((- 7 \sqrt{3}) - (- \sqrt{3}))}^{2}}$

Step 3

Simplify.

Subtract $\sqrt{5}$ from $4 \sqrt{5}$ .

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

Simplify the expression.

Apply the product rule to $3 \sqrt{5}$ .

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

Raise $3$ to the power of $2$ .

$\sqrt{9 {\sqrt{5}}^{2} + {((- 7 \sqrt{3}) - (- \sqrt{3}))}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{9 {(5^{\frac{1}{2}})}^{2} + {((- 7 \sqrt{3}) - (- \sqrt{3}))}^{2}}$

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

$\sqrt{9 \cdot 5^{\frac{1}{2} \cdot 2} + {((- 7 \sqrt{3}) - (- \sqrt{3}))}^{2}}$

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

$\sqrt{9 \cdot 5^{\frac{2}{2}} + {((- 7 \sqrt{3}) - (- \sqrt{3}))}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{9 \cdot 5^{\frac{2}{2}} + {((- 7 \sqrt{3}) - (- \sqrt{3}))}^{2}}$

Rewrite the expression.

$\sqrt{9 \cdot 5^{1} + {((- 7 \sqrt{3}) - (- \sqrt{3}))}^{2}}$

Evaluate the exponent.

$\sqrt{9 \cdot 5 + {((- 7 \sqrt{3}) - (- \sqrt{3}))}^{2}}$

Multiply $9$ by $5$ .

$\sqrt{45 + {((- 7 \sqrt{3}) - (- \sqrt{3}))}^{2}}$

Multiply $- (- \sqrt{3})$ .

Multiply $- 1$ by $- 1$ .

$\sqrt{45 + {(- 7 \sqrt{3} + 1 \sqrt{3})}^{2}}$

Multiply $\sqrt{3}$ by $1$ .

$\sqrt{45 + {(- 7 \sqrt{3} + \sqrt{3})}^{2}}$

Add $- 7 \sqrt{3}$ and $\sqrt{3}$ .

$\sqrt{45 + {(- 6 \sqrt{3})}^{2}}$

Simplify the expression.

Apply the product rule to $- 6 \sqrt{3}$ .

$\sqrt{45 + {(- 6)}^{2} {\sqrt{3}}^{2}}$

Raise $- 6$ to the power of $2$ .

$\sqrt{45 + 36 {\sqrt{3}}^{2}}$

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

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

$\sqrt{45 + 36 {(3^{\frac{1}{2}})}^{2}}$

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

$\sqrt{45 + 36 \cdot 3^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{45 + 36 \cdot 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{45 + 36 \cdot 3^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{45 + 36 \cdot 3^{1}}$

Evaluate the exponent.

$\sqrt{45 + 36 \cdot 3}$

Simplify the expression.

Multiply $36$ by $3$ .

$\sqrt{45 + 108}$

Add $45$ and $108$ .

$\sqrt{153}$

Rewrite $153$ as $3^{2} \cdot 17$ .

Factor $9$ out of $153$ .

$\sqrt{9 (17)}$

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

$\sqrt{3^{2} \cdot 17}$

Pull terms out from under the radical.

$3 \sqrt{17}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt{17}$

Decimal Form:

$12.36931687 \dots$

Step 5

Find the Distance Between Two Points ( square root of 5,- square root of 3) , (4 square root of 5,-7 square root of 3)

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