Basic Math Examples

${(\sqrt{p} + \sqrt{q})}^{2}$

Step 1

Rewrite ${(\sqrt{p} + \sqrt{q})}^{2}$ as $(\sqrt{p} + \sqrt{q}) (\sqrt{p} + \sqrt{q})$ .

$(\sqrt{p} + \sqrt{q}) (\sqrt{p} + \sqrt{q})$

Step 2

Expand $(\sqrt{p} + \sqrt{q}) (\sqrt{p} + \sqrt{q})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\sqrt{p} (\sqrt{p} + \sqrt{q}) + \sqrt{q} (\sqrt{p} + \sqrt{q})$

Step 2.2

Apply the distributive property.

$\sqrt{p} \sqrt{p} + \sqrt{p} \sqrt{q} + \sqrt{q} (\sqrt{p} + \sqrt{q})$

Step 2.3

Apply the distributive property.

$\sqrt{p} \sqrt{p} + \sqrt{p} \sqrt{q} + \sqrt{q} \sqrt{p} + \sqrt{q} \sqrt{q}$

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 $\sqrt{p} \sqrt{p}$ .

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Step 3.1.1.1

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

${\sqrt{p}}^{1} \sqrt{p} + \sqrt{p} \sqrt{q} + \sqrt{q} \sqrt{p} + \sqrt{q} \sqrt{q}$

Step 3.1.1.2

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

${\sqrt{p}}^{1} {\sqrt{p}}^{1} + \sqrt{p} \sqrt{q} + \sqrt{q} \sqrt{p} + \sqrt{q} \sqrt{q}$

Step 3.1.1.3

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

${\sqrt{p}}^{1 + 1} + \sqrt{p} \sqrt{q} + \sqrt{q} \sqrt{p} + \sqrt{q} \sqrt{q}$

Step 3.1.1.4

Add $1$ and $1$ .

${\sqrt{p}}^{2} + \sqrt{p} \sqrt{q} + \sqrt{q} \sqrt{p} + \sqrt{q} \sqrt{q}$

Step 3.1.2

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

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Step 3.1.2.1

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

${(p^{\frac{1}{2}})}^{2} + \sqrt{p} \sqrt{q} + \sqrt{q} \sqrt{p} + \sqrt{q} \sqrt{q}$

Step 3.1.2.2

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

$p^{\frac{1}{2} \cdot 2} + \sqrt{p} \sqrt{q} + \sqrt{q} \sqrt{p} + \sqrt{q} \sqrt{q}$

Step 3.1.2.3

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

$p^{\frac{2}{2}} + \sqrt{p} \sqrt{q} + \sqrt{q} \sqrt{p} + \sqrt{q} \sqrt{q}$

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.

$p^{\frac{2}{2}} + \sqrt{p} \sqrt{q} + \sqrt{q} \sqrt{p} + \sqrt{q} \sqrt{q}$

Step 3.1.2.4.2

Rewrite the expression.

$p^{1} + \sqrt{p} \sqrt{q} + \sqrt{q} \sqrt{p} + \sqrt{q} \sqrt{q}$

Step 3.1.2.5

Simplify.

$p + \sqrt{p} \sqrt{q} + \sqrt{q} \sqrt{p} + \sqrt{q} \sqrt{q}$

Step 3.1.3

Combine using the product rule for radicals.

$p + \sqrt{p q} + \sqrt{q} \sqrt{p} + \sqrt{q} \sqrt{q}$

Step 3.1.4

Combine using the product rule for radicals.

$p + \sqrt{p q} + \sqrt{q p} + \sqrt{q} \sqrt{q}$

Step 3.1.5

Multiply $\sqrt{q} \sqrt{q}$ .

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Step 3.1.5.1

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

$p + \sqrt{p q} + \sqrt{q p} + {\sqrt{q}}^{1} \sqrt{q}$

Step 3.1.5.2

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

$p + \sqrt{p q} + \sqrt{q p} + {\sqrt{q}}^{1} {\sqrt{q}}^{1}$

Step 3.1.5.3

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

$p + \sqrt{p q} + \sqrt{q p} + {\sqrt{q}}^{1 + 1}$

Step 3.1.5.4

Add $1$ and $1$ .

$p + \sqrt{p q} + \sqrt{q p} + {\sqrt{q}}^{2}$

Step 3.1.6

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

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Step 3.1.6.1

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

$p + \sqrt{p q} + \sqrt{q p} + {(q^{\frac{1}{2}})}^{2}$

Step 3.1.6.2

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

$p + \sqrt{p q} + \sqrt{q p} + q^{\frac{1}{2} \cdot 2}$

Step 3.1.6.3

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

$p + \sqrt{p q} + \sqrt{q p} + q^{\frac{2}{2}}$

Step 3.1.6.4

Cancel the common factor of $2$ .

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Step 3.1.6.4.1

Cancel the common factor.

$p + \sqrt{p q} + \sqrt{q p} + q^{\frac{2}{2}}$

Step 3.1.6.4.2

Rewrite the expression.

$p + \sqrt{p q} + \sqrt{q p} + q^{1}$

Step 3.1.6.5

Simplify.

$p + \sqrt{p q} + \sqrt{q p} + q$

Step 3.2

Add $\sqrt{p q}$ and $\sqrt{q p}$ .

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Step 3.2.1

Reorder $q$ and $p$ .

$p + \sqrt{p q} + \sqrt{p q} + q$

Step 3.2.2

Add $\sqrt{p q}$ and $\sqrt{p q}$ .

$p + 2 \sqrt{p q} + q$

Step 4

Move $2 \sqrt{p q}$ .

$p + q + 2 \sqrt{p q}$