Algebra Examples

$(3 \sqrt{2}, 4 \sqrt{3})$ , $(3 \sqrt{2}, - \sqrt{3})$

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.

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

Step 3

Simplify.

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Step 3.1

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

$\sqrt{0^{2} + {((- \sqrt{3}) - 4 \sqrt{3})}^{2}}$

Step 3.2

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

$\sqrt{0 + {((- \sqrt{3}) - 4 \sqrt{3})}^{2}}$

Step 3.3

Subtract $4 \sqrt{3}$ from $- \sqrt{3}$ .

$\sqrt{0 + {(- 5 \sqrt{3})}^{2}}$

Step 3.4

Simplify the expression.

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Step 3.4.1

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

$\sqrt{0 + {(- 5)}^{2} {\sqrt{3}}^{2}}$

Step 3.4.2

Raise $- 5$ to the power of $2$ .

$\sqrt{0 + 25 {\sqrt{3}}^{2}}$

Step 3.5

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

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Step 3.5.1

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

$\sqrt{0 + 25 {(3^{\frac{1}{2}})}^{2}}$

Step 3.5.2

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

$\sqrt{0 + 25 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 3.5.3

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

$\sqrt{0 + 25 \cdot 3^{\frac{2}{2}}}$

Step 3.5.4

Cancel the common factor of $2$ .

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Step 3.5.4.1

Cancel the common factor.

$\sqrt{0 + 25 \cdot 3^{\frac{2}{2}}}$

Step 3.5.4.2

Rewrite the expression.

$\sqrt{0 + 25 \cdot 3^{1}}$

Step 3.5.5

Evaluate the exponent.

$\sqrt{0 + 25 \cdot 3}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Multiply $25$ by $3$ .

$\sqrt{0 + 75}$

Step 3.6.2

Add $0$ and $75$ .

$\sqrt{75}$

Step 3.7

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 3.7.1

Factor $25$ out of $75$ .

$\sqrt{25 (3)}$

Step 3.7.2

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2} \cdot 3}$

Step 3.8

Pull terms out from under the radical.

$5 \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$5 \sqrt{3}$

Decimal Form:

$8.66025403 \dots$

Step 5