Basic Math Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

$\sqrt{y} \sqrt{y} + \sqrt{y} \sqrt{3} + \sqrt{3} \sqrt{y} + \sqrt{3} \sqrt{3}$

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 $\sqrt{y} \sqrt{y}$ .

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Step 3.1.1.1

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

${\sqrt{y}}^{1} \sqrt{y} + \sqrt{y} \sqrt{3} + \sqrt{3} \sqrt{y} + \sqrt{3} \sqrt{3}$

Step 3.1.1.2

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

${\sqrt{y}}^{1} {\sqrt{y}}^{1} + \sqrt{y} \sqrt{3} + \sqrt{3} \sqrt{y} + \sqrt{3} \sqrt{3}$

Step 3.1.1.3

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

${\sqrt{y}}^{1 + 1} + \sqrt{y} \sqrt{3} + \sqrt{3} \sqrt{y} + \sqrt{3} \sqrt{3}$

Step 3.1.1.4

Add $1$ and $1$ .

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

$y^{\frac{1}{2} \cdot 2} + \sqrt{y} \sqrt{3} + \sqrt{3} \sqrt{y} + \sqrt{3} \sqrt{3}$

Step 3.1.2.3

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

$y^{\frac{2}{2}} + \sqrt{y} \sqrt{3} + \sqrt{3} \sqrt{y} + \sqrt{3} \sqrt{3}$

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.

$y^{\frac{2}{2}} + \sqrt{y} \sqrt{3} + \sqrt{3} \sqrt{y} + \sqrt{3} \sqrt{3}$

Step 3.1.2.4.2

Rewrite the expression.

$y^{1} + \sqrt{y} \sqrt{3} + \sqrt{3} \sqrt{y} + \sqrt{3} \sqrt{3}$

Step 3.1.2.5

Simplify.

$y + \sqrt{y} \sqrt{3} + \sqrt{3} \sqrt{y} + \sqrt{3} \sqrt{3}$

Step 3.1.3

Combine using the product rule for radicals.

$y + \sqrt{y \cdot 3} + \sqrt{3} \sqrt{y} + \sqrt{3} \sqrt{3}$

Step 3.1.4

Combine using the product rule for radicals.

$y + \sqrt{y \cdot 3} + \sqrt{3 y} + \sqrt{3} \sqrt{3}$

Step 3.1.5

Combine using the product rule for radicals.

$y + \sqrt{y \cdot 3} + \sqrt{3 y} + \sqrt{3 \cdot 3}$

Step 3.1.6

Multiply $3$ by $3$ .

$y + \sqrt{y \cdot 3} + \sqrt{3 y} + \sqrt{9}$

Step 3.1.7

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

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

Step 3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$y + \sqrt{y \cdot 3} + \sqrt{3 y} + 3$

Step 3.2

Add $\sqrt{y \cdot 3}$ and $\sqrt{3 y}$ .

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Step 3.2.1

Reorder $y$ and $3$ .

$y + \sqrt{3 \cdot y} + \sqrt{3 y} + 3$

Step 3.2.2

Add $\sqrt{3 \cdot y}$ and $\sqrt{3 y}$ .

$y + 2 \sqrt{3 \cdot y} + 3$