Basic Math Examples

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

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.

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

Step 3

Simplify.

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Step 3.1

Subtract $5 \sqrt{3}$ from $- \sqrt{3}$ .

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

Step 3.2

Simplify the expression.

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Step 3.2.1

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

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

Step 3.2.2

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

$\sqrt{36 {\sqrt{3}}^{2} + {(\sqrt{2} - \sqrt{2})}^{2}}$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

$\sqrt{36 \cdot 3^{\frac{2}{2}} + {(\sqrt{2} - \sqrt{2})}^{2}}$

Step 3.3.4

Cancel the common factor of $2$ .

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Step 3.3.4.1

Cancel the common factor.

$\sqrt{36 \cdot 3^{\frac{2}{2}} + {(\sqrt{2} - \sqrt{2})}^{2}}$

Step 3.3.4.2

Rewrite the expression.

$\sqrt{36 \cdot 3^{1} + {(\sqrt{2} - \sqrt{2})}^{2}}$

Step 3.3.5

Evaluate the exponent.

$\sqrt{36 \cdot 3 + {(\sqrt{2} - \sqrt{2})}^{2}}$

Step 3.4

Multiply $36$ by $3$ .

$\sqrt{108 + {(\sqrt{2} - \sqrt{2})}^{2}}$

Step 3.5

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

$\sqrt{108 + 0^{2}}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Raising $0$ to any positive power yields $0$ .

$\sqrt{108 + 0}$

Step 3.6.2

Add $108$ and $0$ .

$\sqrt{108}$

Step 3.7

Rewrite $108$ as $6^{2} \cdot 3$ .

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Step 3.7.1

Factor $36$ out of $108$ .

$\sqrt{36 (3)}$

Step 3.7.2

Rewrite $36$ as $6^{2}$ .

$\sqrt{6^{2} \cdot 3}$

Step 3.8

Pull terms out from under the radical.

$6 \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$6 \sqrt{3}$

Decimal Form:

$10.39230484 \dots$

Step 5