Basic Math Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

$\sqrt{y} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{z} \sqrt{y} + \sqrt{z} \sqrt{z}$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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

Step 2.1.1.4

Add $1$ and $1$ .

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.4.1

Cancel the common factor.

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

Step 2.1.2.4.2

Rewrite the expression.

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

Step 2.1.2.5

Simplify.

$y + \sqrt{y} \sqrt{z} + \sqrt{z} \sqrt{y} + \sqrt{z} \sqrt{z}$

Step 2.1.3

Combine using the product rule for radicals.

$y + \sqrt{y z} + \sqrt{z} \sqrt{y} + \sqrt{z} \sqrt{z}$

Step 2.1.4

Combine using the product rule for radicals.

$y + \sqrt{y z} + \sqrt{z y} + \sqrt{z} \sqrt{z}$

Step 2.1.5

Multiply $\sqrt{z} \sqrt{z}$ .

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Step 2.1.5.1

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

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

Step 2.1.5.2

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

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

Step 2.1.5.3

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

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

Step 2.1.5.4

Add $1$ and $1$ .

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

Step 2.1.6

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

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Step 2.1.6.1

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

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

Step 2.1.6.2

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

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

Step 2.1.6.3

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

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

Step 2.1.6.4

Cancel the common factor of $2$ .

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Step 2.1.6.4.1

Cancel the common factor.

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

Step 2.1.6.4.2

Rewrite the expression.

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

Step 2.1.6.5

Simplify.

$y + \sqrt{y z} + \sqrt{z y} + z$

Step 2.2

Add $\sqrt{y z}$ and $\sqrt{z y}$ .

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Step 2.2.1

Reorder $z$ and $y$ .

$y + \sqrt{y z} + \sqrt{y z} + z$

Step 2.2.2

Add $\sqrt{y z}$ and $\sqrt{y z}$ .

$y + 2 \sqrt{y z} + z$

Step 3

Move $2 \sqrt{y z}$ .

$y + z + 2 \sqrt{y z}$