Find the Distance Between Two Points (2 square root of 3,5 square root of 2) , (-2 square root of 3,- square root of 2)

$(2 \sqrt{3}, 5 \sqrt{2})$ , $(- 2 \sqrt{3}, - \sqrt{2})$

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.

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

Step 3

Simplify.

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

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

Simplify the expression.

Apply the product rule to $- 4 \sqrt{3}$ .

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

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

$\sqrt{16 {\sqrt{3}}^{2} + {((- \sqrt{2}) - 5 \sqrt{2})}^{2}}$

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

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

$\sqrt{16 {(3^{\frac{1}{2}})}^{2} + {((- \sqrt{2}) - 5 \sqrt{2})}^{2}}$

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

$\sqrt{16 \cdot 3^{\frac{1}{2} \cdot 2} + {((- \sqrt{2}) - 5 \sqrt{2})}^{2}}$

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

$\sqrt{16 \cdot 3^{\frac{2}{2}} + {((- \sqrt{2}) - 5 \sqrt{2})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{16 \cdot 3^{\frac{2}{2}} + {((- \sqrt{2}) - 5 \sqrt{2})}^{2}}$

Rewrite the expression.

$\sqrt{16 \cdot 3^{1} + {((- \sqrt{2}) - 5 \sqrt{2})}^{2}}$

Evaluate the exponent.

$\sqrt{16 \cdot 3 + {((- \sqrt{2}) - 5 \sqrt{2})}^{2}}$

Multiply $16$ by $3$ .

$\sqrt{48 + {((- \sqrt{2}) - 5 \sqrt{2})}^{2}}$

Subtract $5 \sqrt{2}$ from $- \sqrt{2}$ .

$\sqrt{48 + {(- 6 \sqrt{2})}^{2}}$

Simplify the expression.

Apply the product rule to $- 6 \sqrt{2}$ .

$\sqrt{48 + {(- 6)}^{2} {\sqrt{2}}^{2}}$

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

$\sqrt{48 + 36 {\sqrt{2}}^{2}}$

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

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

$\sqrt{48 + 36 {(2^{\frac{1}{2}})}^{2}}$

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

$\sqrt{48 + 36 \cdot 2^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{48 + 36 \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{48 + 36 \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{48 + 36 \cdot 2^{1}}$

Evaluate the exponent.

$\sqrt{48 + 36 \cdot 2}$

Simplify the expression.

Multiply $36$ by $2$ .

$\sqrt{48 + 72}$

Add $48$ and $72$ .

$\sqrt{120}$

Rewrite $120$ as $2^{2} \cdot 30$ .

Factor $4$ out of $120$ .

$\sqrt{4 (30)}$

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

$\sqrt{2^{2} \cdot 30}$

Pull terms out from under the radical.

$2 \sqrt{30}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{30}$

Decimal Form:

$10.95445115 \dots$

Step 5

Find the Distance Between Two Points (2 square root of 3,5 square root of 2) , (-2 square root of 3,- square root of 2)

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