Basic Math Examples

$(\sqrt{3} y + \sqrt{3}) (\sqrt{6} y - \sqrt{6})$

Step 1

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

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Step 1.1

Apply the distributive property.

$\sqrt{3} y (\sqrt{6} y - \sqrt{6}) + \sqrt{3} (\sqrt{6} y - \sqrt{6})$

Step 1.2

Apply the distributive property.

$\sqrt{3} y (\sqrt{6} y) + \sqrt{3} y (- \sqrt{6}) + \sqrt{3} (\sqrt{6} y - \sqrt{6})$

Step 1.3

Apply the distributive property.

$\sqrt{3} y (\sqrt{6} y) + \sqrt{3} y (- \sqrt{6}) + \sqrt{3} (\sqrt{6} y) + \sqrt{3} (- \sqrt{6})$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $\sqrt{3} y (\sqrt{6} y) + \sqrt{3} y (- \sqrt{6}) + \sqrt{3} (\sqrt{6} y) + \sqrt{3} (- \sqrt{6})$ .

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Step 2.1.1

Reorder the factors in the terms $\sqrt{3} y (- \sqrt{6})$ and $\sqrt{3} (\sqrt{6} y)$ .

$\sqrt{3} y (\sqrt{6} y) - y \sqrt{3} \sqrt{6} + y \sqrt{3} \sqrt{6} + \sqrt{3} (- \sqrt{6})$

Step 2.1.2

Add $- y \sqrt{3} \sqrt{6}$ and $y \sqrt{3} \sqrt{6}$ .

$\sqrt{3} y (\sqrt{6} y) + 0 + \sqrt{3} (- \sqrt{6})$

Step 2.1.3

Add $\sqrt{3} y (\sqrt{6} y)$ and $0$ .

$\sqrt{3} y (\sqrt{6} y) + \sqrt{3} (- \sqrt{6})$

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite using the commutative property of multiplication.

$\sqrt{3} \cdot \sqrt{6} y \cdot y + \sqrt{3} (- \sqrt{6})$

Step 2.2.2

Multiply $y$ by $y$ by adding the exponents.

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Step 2.2.2.1

Move $y$ .

$\sqrt{3} \cdot \sqrt{6} (y \cdot y) + \sqrt{3} (- \sqrt{6})$

Step 2.2.2.2

Multiply $y$ by $y$ .

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

Step 2.2.3

Combine using the product rule for radicals.

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

Step 2.2.4

Multiply $3$ by $6$ .

$\sqrt{18} y^{2} + \sqrt{3} (- \sqrt{6})$

Step 2.2.5

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

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Step 2.2.5.1

Factor $9$ out of $18$ .

$\sqrt{9 (2)} y^{2} + \sqrt{3} (- \sqrt{6})$

Step 2.2.5.2

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

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

Step 2.2.6

Pull terms out from under the radical.

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

Step 2.2.7

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

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Step 2.2.7.1

Combine using the product rule for radicals.

$3 \sqrt{2} y^{2} - \sqrt{3 \cdot 6}$

Step 2.2.7.2

Multiply $3$ by $6$ .

$3 \sqrt{2} y^{2} - \sqrt{18}$

Step 2.2.8

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

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Step 2.2.8.1

Factor $9$ out of $18$ .

$3 \sqrt{2} y^{2} - \sqrt{9 (2)}$

Step 2.2.8.2

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

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

Step 2.2.9

Pull terms out from under the radical.

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

Step 2.2.10

Multiply $3$ by $- 1$ .

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