Precalculus Examples

$\sqrt{18} - \sqrt{50} + \sqrt{12} - \sqrt{75}$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $9$ out of $18$ .

$\sqrt{9 (2)} - \sqrt{50} + \sqrt{12} - \sqrt{75}$

Step 1.1.2

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

$\sqrt{3^{2} \cdot 2} - \sqrt{50} + \sqrt{12} - \sqrt{75}$

Step 1.2

Pull terms out from under the radical.

$3 \sqrt{2} - \sqrt{50} + \sqrt{12} - \sqrt{75}$

Step 1.3

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 1.3.1

Factor $25$ out of $50$ .

$3 \sqrt{2} - \sqrt{25 (2)} + \sqrt{12} - \sqrt{75}$

Step 1.3.2

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

$3 \sqrt{2} - \sqrt{5^{2} \cdot 2} + \sqrt{12} - \sqrt{75}$

Step 1.4

Pull terms out from under the radical.

$3 \sqrt{2} - (5 \sqrt{2}) + \sqrt{12} - \sqrt{75}$

Step 1.5

Multiply $5$ by $- 1$ .

$3 \sqrt{2} - 5 \sqrt{2} + \sqrt{12} - \sqrt{75}$

Step 1.6

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

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Step 1.6.1

Factor $4$ out of $12$ .

$3 \sqrt{2} - 5 \sqrt{2} + \sqrt{4 (3)} - \sqrt{75}$

Step 1.6.2

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

$3 \sqrt{2} - 5 \sqrt{2} + \sqrt{2^{2} \cdot 3} - \sqrt{75}$

Step 1.7

Pull terms out from under the radical.

$3 \sqrt{2} - 5 \sqrt{2} + 2 \sqrt{3} - \sqrt{75}$

Step 1.8

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

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Step 1.8.1

Factor $25$ out of $75$ .

$3 \sqrt{2} - 5 \sqrt{2} + 2 \sqrt{3} - \sqrt{25 (3)}$

Step 1.8.2

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

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

Step 1.9

Pull terms out from under the radical.

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

Step 1.10

Multiply $5$ by $- 1$ .

$3 \sqrt{2} - 5 \sqrt{2} + 2 \sqrt{3} - 5 \sqrt{3}$

Step 2

Simplify by adding terms.

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Step 2.1

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

$- 2 \sqrt{2} + 2 \sqrt{3} - 5 \sqrt{3}$

Step 2.2

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

$- 2 \sqrt{2} - 3 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 2 \sqrt{2} - 3 \sqrt{3}$

Decimal Form:

$- 8.02457954 \dots$