Algebra Examples

$2 \sqrt{45} - 4 \sqrt{135} - 2 \sqrt{80} + 2 \sqrt{60}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 1.1.1

Factor $9$ out of $45$ .

$2 \sqrt{9 (5)} - 4 \sqrt{135} - 2 \sqrt{80} + 2 \sqrt{60}$

Step 1.1.2

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

$2 \sqrt{3^{2} \cdot 5} - 4 \sqrt{135} - 2 \sqrt{80} + 2 \sqrt{60}$

Step 1.2

Pull terms out from under the radical.

$2 (| 3 | \sqrt{5}) - 4 \sqrt{135} - 2 \sqrt{80} + 2 \sqrt{60}$

Step 1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$2 (3 \sqrt{5}) - 4 \sqrt{135} - 2 \sqrt{80} + 2 \sqrt{60}$

Step 1.4

Multiply $3$ by $2$ .

$6 \sqrt{5} - 4 \sqrt{135} - 2 \sqrt{80} + 2 \sqrt{60}$

Step 1.5

Rewrite $135$ as $3^{2} \cdot 15$ .

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Step 1.5.1

Factor $9$ out of $135$ .

$6 \sqrt{5} - 4 \sqrt{9 (15)} - 2 \sqrt{80} + 2 \sqrt{60}$

Step 1.5.2

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

$6 \sqrt{5} - 4 \sqrt{3^{2} \cdot 15} - 2 \sqrt{80} + 2 \sqrt{60}$

Step 1.6

Pull terms out from under the radical.

$6 \sqrt{5} - 4 (| 3 | \sqrt{15}) - 2 \sqrt{80} + 2 \sqrt{60}$

Step 1.7

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$6 \sqrt{5} - 4 (3 \sqrt{15}) - 2 \sqrt{80} + 2 \sqrt{60}$

Step 1.8

Multiply $3$ by $- 4$ .

$6 \sqrt{5} - 12 \sqrt{15} - 2 \sqrt{80} + 2 \sqrt{60}$

Step 1.9

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 1.9.1

Factor $16$ out of $80$ .

$6 \sqrt{5} - 12 \sqrt{15} - 2 \sqrt{16 (5)} + 2 \sqrt{60}$

Step 1.9.2

Rewrite $16$ as $4^{2}$ .

$6 \sqrt{5} - 12 \sqrt{15} - 2 \sqrt{4^{2} \cdot 5} + 2 \sqrt{60}$

Step 1.10

Pull terms out from under the radical.

$6 \sqrt{5} - 12 \sqrt{15} - 2 (| 4 | \sqrt{5}) + 2 \sqrt{60}$

Step 1.11

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$6 \sqrt{5} - 12 \sqrt{15} - 2 (4 \sqrt{5}) + 2 \sqrt{60}$

Step 1.12

Multiply $4$ by $- 2$ .

$6 \sqrt{5} - 12 \sqrt{15} - 8 \sqrt{5} + 2 \sqrt{60}$

Step 1.13

Rewrite $60$ as $2^{2} \cdot 15$ .

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Step 1.13.1

Factor $4$ out of $60$ .

$6 \sqrt{5} - 12 \sqrt{15} - 8 \sqrt{5} + 2 \sqrt{4 (15)}$

Step 1.13.2

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

$6 \sqrt{5} - 12 \sqrt{15} - 8 \sqrt{5} + 2 \sqrt{2^{2} \cdot 15}$

Step 1.14

Pull terms out from under the radical.

$6 \sqrt{5} - 12 \sqrt{15} - 8 \sqrt{5} + 2 (| 2 | \sqrt{15})$

Step 1.15

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$6 \sqrt{5} - 12 \sqrt{15} - 8 \sqrt{5} + 2 (2 \sqrt{15})$

Step 1.16

Multiply $2$ by $2$ .

$6 \sqrt{5} - 12 \sqrt{15} - 8 \sqrt{5} + 4 \sqrt{15}$

Step 2

Subtract $8 \sqrt{5}$ from $6 \sqrt{5}$ .

$- 2 \sqrt{5} - 12 \sqrt{15} + 4 \sqrt{15}$

Step 3

Add $- 12 \sqrt{15}$ and $4 \sqrt{15}$ .

$- 2 \sqrt{5} - 8 \sqrt{15}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- 2 \sqrt{5} - 8 \sqrt{15}$

Decimal Form:

$- 35.45600272 \dots$