Basic Math Examples

${(\sqrt{12 y} - \sqrt{6 y})}^{2}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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Step 1.1.1.1

Factor $4$ out of $12$ .

${(\sqrt{4 (3) y} - \sqrt{6 y})}^{2}$

Step 1.1.1.2

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

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

Step 1.1.1.3

Add parentheses.

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

Step 1.1.2

Pull terms out from under the radical.

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

Step 1.2

Rewrite ${(2 \sqrt{3 y} - \sqrt{6 y})}^{2}$ as $(2 \sqrt{3 y} - \sqrt{6 y}) (2 \sqrt{3 y} - \sqrt{6 y})$ .

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $2 \sqrt{3 y} (2 \sqrt{3 y})$ .

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Step 3.1.1.1

Multiply $2$ by $2$ .

$4 \sqrt{3 y} \sqrt{3 y} + 2 \sqrt{3 y} (- \sqrt{6 y}) - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.1.2

Raise $\sqrt{3 y}$ to the power of $1$ .

$4 ({\sqrt{3 y}}^{1} \sqrt{3 y}) + 2 \sqrt{3 y} (- \sqrt{6 y}) - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.1.3

Raise $\sqrt{3 y}$ to the power of $1$ .

$4 ({\sqrt{3 y}}^{1} {\sqrt{3 y}}^{1}) + 2 \sqrt{3 y} (- \sqrt{6 y}) - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 {\sqrt{3 y}}^{1 + 1} + 2 \sqrt{3 y} (- \sqrt{6 y}) - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.1.5

Add $1$ and $1$ .

$4 {\sqrt{3 y}}^{2} + 2 \sqrt{3 y} (- \sqrt{6 y}) - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.2

Rewrite ${\sqrt{3 y}}^{2}$ as $3 y$ .

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Step 3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 y}$ as ${(3 y)}^{\frac{1}{2}}$ .

$4 {({(3 y)}^{\frac{1}{2}})}^{2} + 2 \sqrt{3 y} (- \sqrt{6 y}) - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.2.2

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

$4 {(3 y)}^{\frac{1}{2} \cdot 2} + 2 \sqrt{3 y} (- \sqrt{6 y}) - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.2.3

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

$4 {(3 y)}^{\frac{2}{2}} + 2 \sqrt{3 y} (- \sqrt{6 y}) - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.2.4

Cancel the common factor of $2$ .

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Step 3.1.2.4.1

Cancel the common factor.

$4 {(3 y)}^{\frac{2}{2}} + 2 \sqrt{3 y} (- \sqrt{6 y}) - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.2.4.2

Rewrite the expression.

$4 {(3 y)}^{1} + 2 \sqrt{3 y} (- \sqrt{6 y}) - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.2.5

Simplify.

$4 (3 y) + 2 \sqrt{3 y} (- \sqrt{6 y}) - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.3

Multiply $3$ by $4$ .

$12 y + 2 \sqrt{3 y} (- \sqrt{6 y}) - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.4

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

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Step 3.1.4.1

Multiply $- 1$ by $2$ .

$12 y - 2 \sqrt{3 y} \sqrt{6 y} - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.4.2

Combine using the product rule for radicals.

$12 y - 2 \sqrt{6 y (3 y)} - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.4.3

Multiply $3$ by $6$ .

$12 y - 2 \sqrt{18 y \cdot y} - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.4.4

Raise $y$ to the power of $1$ .

$12 y - 2 \sqrt{18 (y^{1} y)} - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.4.5

Raise $y$ to the power of $1$ .

$12 y - 2 \sqrt{18 (y^{1} y^{1})} - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.4.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 y - 2 \sqrt{18 y^{1 + 1}} - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.4.7

Add $1$ and $1$ .

$12 y - 2 \sqrt{18 y^{2}} - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.5

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

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Step 3.1.5.1

Factor $9$ out of $18$ .

$12 y - 2 \sqrt{9 (2) y^{2}} - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.5.2

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

$12 y - 2 \sqrt{3^{2} \cdot 2 y^{2}} - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.5.3

Move $2$ .

$12 y - 2 \sqrt{3^{2} y^{2} \cdot 2} - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.5.4

Rewrite $3^{2} y^{2}$ as ${(3 y)}^{2}$ .

$12 y - 2 \sqrt{{(3 y)}^{2} \cdot 2} - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.6

Pull terms out from under the radical.

$12 y - 2 (3 y \sqrt{2}) - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.7

Multiply $3$ by $- 2$ .

$12 y - 6 (y \sqrt{2}) - \sqrt{6 y} (2 \sqrt{3 y}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.8

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

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Step 3.1.8.1

Multiply $2$ by $- 1$ .

$12 y - 6 y \sqrt{2} - 2 \sqrt{6 y} \sqrt{3 y} - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.8.2

Combine using the product rule for radicals.

$12 y - 6 y \sqrt{2} - 2 \sqrt{3 y (6 y)} - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.8.3

Multiply $6$ by $3$ .

$12 y - 6 y \sqrt{2} - 2 \sqrt{18 y \cdot y} - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.8.4

Raise $y$ to the power of $1$ .

$12 y - 6 y \sqrt{2} - 2 \sqrt{18 (y^{1} y)} - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.8.5

Raise $y$ to the power of $1$ .

$12 y - 6 y \sqrt{2} - 2 \sqrt{18 (y^{1} y^{1})} - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.8.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 y - 6 y \sqrt{2} - 2 \sqrt{18 y^{1 + 1}} - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.8.7

Add $1$ and $1$ .

$12 y - 6 y \sqrt{2} - 2 \sqrt{18 y^{2}} - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.9

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

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Step 3.1.9.1

Factor $9$ out of $18$ .

$12 y - 6 y \sqrt{2} - 2 \sqrt{9 (2) y^{2}} - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.9.2

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

$12 y - 6 y \sqrt{2} - 2 \sqrt{3^{2} \cdot 2 y^{2}} - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.9.3

Move $2$ .

$12 y - 6 y \sqrt{2} - 2 \sqrt{3^{2} y^{2} \cdot 2} - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.9.4

Rewrite $3^{2} y^{2}$ as ${(3 y)}^{2}$ .

$12 y - 6 y \sqrt{2} - 2 \sqrt{{(3 y)}^{2} \cdot 2} - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.10

Pull terms out from under the radical.

$12 y - 6 y \sqrt{2} - 2 (3 y \sqrt{2}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.11

Multiply $3$ by $- 2$ .

$12 y - 6 y \sqrt{2} - 6 (y \sqrt{2}) - \sqrt{6 y} (- \sqrt{6 y})$

Step 3.1.12

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

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Step 3.1.12.1

Multiply $- 1$ by $- 1$ .

$12 y - 6 y \sqrt{2} - 6 y \sqrt{2} + 1 \sqrt{6 y} \sqrt{6 y}$

Step 3.1.12.2

Multiply $\sqrt{6 y}$ by $1$ .

$12 y - 6 y \sqrt{2} - 6 y \sqrt{2} + \sqrt{6 y} \sqrt{6 y}$

Step 3.1.12.3

Raise $\sqrt{6 y}$ to the power of $1$ .

$12 y - 6 y \sqrt{2} - 6 y \sqrt{2} + {\sqrt{6 y}}^{1} \sqrt{6 y}$

Step 3.1.12.4

Raise $\sqrt{6 y}$ to the power of $1$ .

$12 y - 6 y \sqrt{2} - 6 y \sqrt{2} + {\sqrt{6 y}}^{1} {\sqrt{6 y}}^{1}$

Step 3.1.12.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 y - 6 y \sqrt{2} - 6 y \sqrt{2} + {\sqrt{6 y}}^{1 + 1}$

Step 3.1.12.6

Add $1$ and $1$ .

$12 y - 6 y \sqrt{2} - 6 y \sqrt{2} + {\sqrt{6 y}}^{2}$

Step 3.1.13

Rewrite ${\sqrt{6 y}}^{2}$ as $6 y$ .

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Step 3.1.13.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 y}$ as ${(6 y)}^{\frac{1}{2}}$ .

$12 y - 6 y \sqrt{2} - 6 y \sqrt{2} + {({(6 y)}^{\frac{1}{2}})}^{2}$

Step 3.1.13.2

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

$12 y - 6 y \sqrt{2} - 6 y \sqrt{2} + {(6 y)}^{\frac{1}{2} \cdot 2}$

Step 3.1.13.3

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

$12 y - 6 y \sqrt{2} - 6 y \sqrt{2} + {(6 y)}^{\frac{2}{2}}$

Step 3.1.13.4

Cancel the common factor of $2$ .

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Step 3.1.13.4.1

Cancel the common factor.

$12 y - 6 y \sqrt{2} - 6 y \sqrt{2} + {(6 y)}^{\frac{2}{2}}$

Step 3.1.13.4.2

Rewrite the expression.

$12 y - 6 y \sqrt{2} - 6 y \sqrt{2} + {(6 y)}^{1}$

Step 3.1.13.5

Simplify.

$12 y - 6 y \sqrt{2} - 6 y \sqrt{2} + 6 y$

Step 3.2

Add $12 y$ and $6 y$ .

$- 6 y \sqrt{2} - 6 y \sqrt{2} + 18 y$

Step 3.3

Subtract $6 y \sqrt{2}$ from $- 6 y \sqrt{2}$ .

$- 12 y \sqrt{2} + 18 y$