Algebra Examples

$(\sqrt{10} + \sqrt{6}) (\sqrt{30} - \sqrt{18})$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Factor $9$ out of $18$ .

$(\sqrt{10} + \sqrt{6}) (\sqrt{30} - \sqrt{9 (2)})$

Step 1.1.2

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

$(\sqrt{10} + \sqrt{6}) (\sqrt{30} - \sqrt{3^{2} \cdot 2})$

Step 1.2

Pull terms out from under the radical.

$(\sqrt{10} + \sqrt{6}) (\sqrt{30} - (3 \sqrt{2}))$

Step 1.3

Multiply $3$ by $- 1$ .

$(\sqrt{10} + \sqrt{6}) (\sqrt{30} - 3 \sqrt{2})$

Step 2

Expand $(\sqrt{10} + \sqrt{6}) (\sqrt{30} - 3 \sqrt{2})$ using the FOIL Method.

Tap for more steps...

Step 2.1

Apply the distributive property.

$\sqrt{10} (\sqrt{30} - 3 \sqrt{2}) + \sqrt{6} (\sqrt{30} - 3 \sqrt{2})$

Step 2.2

Apply the distributive property.

$\sqrt{10} \sqrt{30} + \sqrt{10} (- 3 \sqrt{2}) + \sqrt{6} (\sqrt{30} - 3 \sqrt{2})$

Step 2.3

Apply the distributive property.

$\sqrt{10} \sqrt{30} + \sqrt{10} (- 3 \sqrt{2}) + \sqrt{6} \sqrt{30} + \sqrt{6} (- 3 \sqrt{2})$

Step 3

Simplify and combine like terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Combine using the product rule for radicals.

$\sqrt{10 \cdot 30} + \sqrt{10} (- 3 \sqrt{2}) + \sqrt{6} \sqrt{30} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.2

Multiply $10$ by $30$ .

$\sqrt{300} + \sqrt{10} (- 3 \sqrt{2}) + \sqrt{6} \sqrt{30} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.3

Rewrite $300$ as $10^{2} \cdot 3$ .

Tap for more steps...

Step 3.1.3.1

Factor $100$ out of $300$ .

$\sqrt{100 (3)} + \sqrt{10} (- 3 \sqrt{2}) + \sqrt{6} \sqrt{30} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.3.2

Rewrite $100$ as $10^{2}$ .

$\sqrt{10^{2} \cdot 3} + \sqrt{10} (- 3 \sqrt{2}) + \sqrt{6} \sqrt{30} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.4

Pull terms out from under the radical.

$10 \sqrt{3} + \sqrt{10} (- 3 \sqrt{2}) + \sqrt{6} \sqrt{30} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.5

Multiply $\sqrt{10} (- 3 \sqrt{2})$ .

Tap for more steps...

Step 3.1.5.1

Combine using the product rule for radicals.

$10 \sqrt{3} - 3 \sqrt{10 \cdot 2} + \sqrt{6} \sqrt{30} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.5.2

Multiply $10$ by $2$ .

$10 \sqrt{3} - 3 \sqrt{20} + \sqrt{6} \sqrt{30} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.6

Rewrite $20$ as $2^{2} \cdot 5$ .

Tap for more steps...

Step 3.1.6.1

Factor $4$ out of $20$ .

$10 \sqrt{3} - 3 \sqrt{4 (5)} + \sqrt{6} \sqrt{30} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.6.2

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

$10 \sqrt{3} - 3 \sqrt{2^{2} \cdot 5} + \sqrt{6} \sqrt{30} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.7

Pull terms out from under the radical.

$10 \sqrt{3} - 3 (2 \sqrt{5}) + \sqrt{6} \sqrt{30} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.8

Multiply $2$ by $- 3$ .

$10 \sqrt{3} - 6 \sqrt{5} + \sqrt{6} \sqrt{30} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.9

Combine using the product rule for radicals.

$10 \sqrt{3} - 6 \sqrt{5} + \sqrt{6 \cdot 30} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.10

Multiply $6$ by $30$ .

$10 \sqrt{3} - 6 \sqrt{5} + \sqrt{180} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.11

Rewrite $180$ as $6^{2} \cdot 5$ .

Tap for more steps...

Step 3.1.11.1

Factor $36$ out of $180$ .

$10 \sqrt{3} - 6 \sqrt{5} + \sqrt{36 (5)} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.11.2

Rewrite $36$ as $6^{2}$ .

$10 \sqrt{3} - 6 \sqrt{5} + \sqrt{6^{2} \cdot 5} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.12

Pull terms out from under the radical.

$10 \sqrt{3} - 6 \sqrt{5} + 6 \sqrt{5} + \sqrt{6} (- 3 \sqrt{2})$

Step 3.1.13

Multiply $\sqrt{6} (- 3 \sqrt{2})$ .

Tap for more steps...

Step 3.1.13.1

Combine using the product rule for radicals.

$10 \sqrt{3} - 6 \sqrt{5} + 6 \sqrt{5} - 3 \sqrt{6 \cdot 2}$

Step 3.1.13.2

Multiply $6$ by $2$ .

$10 \sqrt{3} - 6 \sqrt{5} + 6 \sqrt{5} - 3 \sqrt{12}$

Step 3.1.14

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

Tap for more steps...

Step 3.1.14.1

Factor $4$ out of $12$ .

$10 \sqrt{3} - 6 \sqrt{5} + 6 \sqrt{5} - 3 \sqrt{4 (3)}$

Step 3.1.14.2

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

$10 \sqrt{3} - 6 \sqrt{5} + 6 \sqrt{5} - 3 \sqrt{2^{2} \cdot 3}$

Step 3.1.15

Pull terms out from under the radical.

$10 \sqrt{3} - 6 \sqrt{5} + 6 \sqrt{5} - 3 (2 \sqrt{3})$

Step 3.1.16

Multiply $2$ by $- 3$ .

$10 \sqrt{3} - 6 \sqrt{5} + 6 \sqrt{5} - 6 \sqrt{3}$

Step 3.2

Subtract $6 \sqrt{3}$ from $10 \sqrt{3}$ .

$4 \sqrt{3} - 6 \sqrt{5} + 6 \sqrt{5}$

Step 3.3

Add $- 6 \sqrt{5}$ and $6 \sqrt{5}$ .

$4 \sqrt{3} + 0$

Step 3.4

Add $4 \sqrt{3}$ and $0$ .

$4 \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{3}$

Decimal Form:

$6.92820323 \dots$