Algebra Examples

$(6, 5 \sqrt{2})$ , $(4, 3 \sqrt{2})$

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.

$\sqrt{{(4 - 6)}^{2} + {(3 \sqrt{2} - 5 \sqrt{2})}^{2}}$

Step 3

Simplify.

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Step 3.1

Simplify the expression.

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Step 3.1.1

Subtract $6$ from $4$ .

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

Step 3.1.2

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

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

Step 3.2

Subtract $5 \sqrt{2}$ from $3 \sqrt{2}$ .

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

Step 3.3

Simplify the expression.

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Step 3.3.1

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

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

Step 3.3.2

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

$\sqrt{4 + 4 {\sqrt{2}}^{2}}$

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

$\sqrt{4 + 4 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 3.4.3

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

$\sqrt{4 + 4 \cdot 2^{\frac{2}{2}}}$

Step 3.4.4

Cancel the common factor of $2$ .

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Step 3.4.4.1

Cancel the common factor.

$\sqrt{4 + 4 \cdot 2^{\frac{2}{2}}}$

Step 3.4.4.2

Rewrite the expression.

$\sqrt{4 + 4 \cdot 2^{1}}$

Step 3.4.5

Evaluate the exponent.

$\sqrt{4 + 4 \cdot 2}$

Step 3.5

Simplify the expression.

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Step 3.5.1

Multiply $4$ by $2$ .

$\sqrt{4 + 8}$

Step 3.5.2

Add $4$ and $8$ .

$\sqrt{12}$

Step 3.6

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 3.6.1

Factor $4$ out of $12$ .

$\sqrt{4 (3)}$

Step 3.6.2

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

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

Step 3.7

Pull terms out from under the radical.

$2 \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{3}$

Decimal Form:

$3.46410161 \dots$

Step 5