Basic Math Examples

$(- 4 \sqrt{2}, - \sqrt{8})$ , $(- 5 \sqrt{2}, \sqrt{18})$

Step 1

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

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Step 1.1

Factor $4$ out of $8$ .

$(- 4 \sqrt{2}, - \sqrt{4 (2)}) - (- 5 \sqrt{2}, \sqrt{18})$

Step 1.2

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

$(- 4 \sqrt{2}, - \sqrt{2^{2} \cdot 2}) - (- 5 \sqrt{2}, \sqrt{18})$

Step 2

Pull terms out from under the radical.

$(- 4 \sqrt{2}, - (2 \sqrt{2})) - (- 5 \sqrt{2}, \sqrt{18})$

Step 3

Multiply $2$ by $- 1$ .

$(- 4 \sqrt{2}, - 2 \sqrt{2}) - (- 5 \sqrt{2}, \sqrt{18})$

Step 4

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 4.1

Factor $9$ out of $18$ .

$(- 4 \sqrt{2}, - 2 \sqrt{2}) - (- 5 \sqrt{2}, \sqrt{9 (2)})$

Step 4.2

Rewrite $9$ as $3^{2}$ .

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

Step 5

Pull terms out from under the radical.

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

Step 6

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 7

Substitute the actual values of the points into the distance formula.

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

Step 8

Simplify.

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Step 8.1

Multiply $- 4$ by $- 1$ .

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

Step 8.2

Add $- 5 \sqrt{2}$ and $4 \sqrt{2}$ .

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

Step 8.3

Simplify the expression.

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Step 8.3.1

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

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

Step 8.3.2

Raise $- 1$ to the power of $2$ .

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

Step 8.3.3

Multiply ${\sqrt{2}}^{2}$ by $1$ .

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

Step 8.4

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

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Step 8.4.1

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

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

Step 8.4.2

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

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

Step 8.4.3

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

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

Step 8.4.4

Cancel the common factor of $2$ .

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Step 8.4.4.1

Cancel the common factor.

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

Step 8.4.4.2

Rewrite the expression.

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

Step 8.4.5

Evaluate the exponent.

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

Step 8.5

Multiply $- 2$ by $- 1$ .

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

Step 8.6

Add $3 \sqrt{2}$ and $2 \sqrt{2}$ .

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

Step 8.7

Simplify the expression.

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Step 8.7.1

Apply the product rule to $5 \sqrt{2}$ .

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

Step 8.7.2

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

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

Step 8.8

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

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Step 8.8.1

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

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

Step 8.8.2

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

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

Step 8.8.3

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

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

Step 8.8.4

Cancel the common factor of $2$ .

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Step 8.8.4.1

Cancel the common factor.

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

Step 8.8.4.2

Rewrite the expression.

$\sqrt{2 + 25 \cdot 2^{1}}$

Step 8.8.5

Evaluate the exponent.

$\sqrt{2 + 25 \cdot 2}$

Step 8.9

Simplify the expression.

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Step 8.9.1

Multiply $25$ by $2$ .

$\sqrt{2 + 50}$

Step 8.9.2

Add $2$ and $50$ .

$\sqrt{52}$

Step 8.10

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 8.10.1

Factor $4$ out of $52$ .

$\sqrt{4 (13)}$

Step 8.10.2

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

$\sqrt{2^{2} \cdot 13}$

Step 8.11

Pull terms out from under the radical.

$2 \sqrt{13}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{13}$

Decimal Form:

$7.21110255 \dots$

Step 10