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

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

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.

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

Step 3

Simplify.

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

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

Simplify the expression.

Apply the product rule to $- 2 \sqrt{3}$ .

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

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

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

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

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

$\sqrt{4 {(3^{\frac{1}{2}})}^{2} + {((- \sqrt{5}) - 5 \sqrt{5})}^{2}}$

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

$\sqrt{4 \cdot 3^{\frac{1}{2} \cdot 2} + {((- \sqrt{5}) - 5 \sqrt{5})}^{2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\sqrt{4 \cdot 3^{1} + {((- \sqrt{5}) - 5 \sqrt{5})}^{2}}$

Evaluate the exponent.

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

Multiply $4$ by $3$ .

$\sqrt{12 + {((- \sqrt{5}) - 5 \sqrt{5})}^{2}}$

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

$\sqrt{12 + {(- 6 \sqrt{5})}^{2}}$

Simplify the expression.

Apply the product rule to $- 6 \sqrt{5}$ .

$\sqrt{12 + {(- 6)}^{2} {\sqrt{5}}^{2}}$

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

$\sqrt{12 + 36 {\sqrt{5}}^{2}}$

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

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

$\sqrt{12 + 36 {(5^{\frac{1}{2}})}^{2}}$

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

$\sqrt{12 + 36 \cdot 5^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{12 + 36 \cdot 5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{12 + 36 \cdot 5^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{12 + 36 \cdot 5^{1}}$

Evaluate the exponent.

$\sqrt{12 + 36 \cdot 5}$

Simplify the expression.

Multiply $36$ by $5$ .

$\sqrt{12 + 180}$

Add $12$ and $180$ .

$\sqrt{192}$

Rewrite $192$ as $8^{2} \cdot 3$ .

Factor $64$ out of $192$ .

$\sqrt{64 (3)}$

Rewrite $64$ as $8^{2}$ .

$\sqrt{8^{2} \cdot 3}$

Pull terms out from under the radical.

$8 \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$8 \sqrt{3}$

Decimal Form:

$13.85640646 \dots$

Step 5

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

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