Basic Math Examples

$(3 \sqrt{p} + \sqrt{q}) (3 \sqrt{p} - \sqrt{q})$

Step 1

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

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Step 1.1

Apply the distributive property.

$3 \sqrt{p} (3 \sqrt{p} - \sqrt{q}) + \sqrt{q} (3 \sqrt{p} - \sqrt{q})$

Step 1.2

Apply the distributive property.

$3 \sqrt{p} (3 \sqrt{p}) + 3 \sqrt{p} (- \sqrt{q}) + \sqrt{q} (3 \sqrt{p} - \sqrt{q})$

Step 1.3

Apply the distributive property.

$3 \sqrt{p} (3 \sqrt{p}) + 3 \sqrt{p} (- \sqrt{q}) + \sqrt{q} (3 \sqrt{p}) + \sqrt{q} (- \sqrt{q})$

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 $3 \sqrt{p} (3 \sqrt{p})$ .

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Step 2.1.1.1

Multiply $3$ by $3$ .

$9 \sqrt{p} \sqrt{p} + 3 \sqrt{p} (- \sqrt{q}) + \sqrt{q} (3 \sqrt{p}) + \sqrt{q} (- \sqrt{q})$

Step 2.1.1.2

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

$9 ({\sqrt{p}}^{1} \sqrt{p}) + 3 \sqrt{p} (- \sqrt{q}) + \sqrt{q} (3 \sqrt{p}) + \sqrt{q} (- \sqrt{q})$

Step 2.1.1.3

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

$9 ({\sqrt{p}}^{1} {\sqrt{p}}^{1}) + 3 \sqrt{p} (- \sqrt{q}) + \sqrt{q} (3 \sqrt{p}) + \sqrt{q} (- \sqrt{q})$

Step 2.1.1.4

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

$9 {\sqrt{p}}^{1 + 1} + 3 \sqrt{p} (- \sqrt{q}) + \sqrt{q} (3 \sqrt{p}) + \sqrt{q} (- \sqrt{q})$

Step 2.1.1.5

Add $1$ and $1$ .

$9 {\sqrt{p}}^{2} + 3 \sqrt{p} (- \sqrt{q}) + \sqrt{q} (3 \sqrt{p}) + \sqrt{q} (- \sqrt{q})$

Step 2.1.2

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

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Step 2.1.2.1

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

$9 {(p^{\frac{1}{2}})}^{2} + 3 \sqrt{p} (- \sqrt{q}) + \sqrt{q} (3 \sqrt{p}) + \sqrt{q} (- \sqrt{q})$

Step 2.1.2.2

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

$9 p^{\frac{1}{2} \cdot 2} + 3 \sqrt{p} (- \sqrt{q}) + \sqrt{q} (3 \sqrt{p}) + \sqrt{q} (- \sqrt{q})$

Step 2.1.2.3

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

$9 p^{\frac{2}{2}} + 3 \sqrt{p} (- \sqrt{q}) + \sqrt{q} (3 \sqrt{p}) + \sqrt{q} (- \sqrt{q})$

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.

$9 p^{\frac{2}{2}} + 3 \sqrt{p} (- \sqrt{q}) + \sqrt{q} (3 \sqrt{p}) + \sqrt{q} (- \sqrt{q})$

Step 2.1.2.4.2

Rewrite the expression.

$9 p^{1} + 3 \sqrt{p} (- \sqrt{q}) + \sqrt{q} (3 \sqrt{p}) + \sqrt{q} (- \sqrt{q})$

Step 2.1.2.5

Simplify.

$9 p + 3 \sqrt{p} (- \sqrt{q}) + \sqrt{q} (3 \sqrt{p}) + \sqrt{q} (- \sqrt{q})$

Step 2.1.3

Multiply $3 \sqrt{p} (- \sqrt{q})$ .

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Step 2.1.3.1

Multiply $- 1$ by $3$ .

$9 p - 3 \sqrt{p} \sqrt{q} + \sqrt{q} (3 \sqrt{p}) + \sqrt{q} (- \sqrt{q})$

Step 2.1.3.2

Combine using the product rule for radicals.

$9 p - 3 \sqrt{q p} + \sqrt{q} (3 \sqrt{p}) + \sqrt{q} (- \sqrt{q})$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$9 p - 3 \sqrt{q p} + 3 \sqrt{q} \sqrt{p} + \sqrt{q} (- \sqrt{q})$

Step 2.1.5

Combine using the product rule for radicals.

$9 p - 3 \sqrt{q p} + 3 \sqrt{p q} + \sqrt{q} (- \sqrt{q})$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$9 p - 3 \sqrt{q p} + 3 \sqrt{p q} - \sqrt{q} \sqrt{q}$

Step 2.1.7

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

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Step 2.1.7.1

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

$9 p - 3 \sqrt{q p} + 3 \sqrt{p q} - ({\sqrt{q}}^{1} \sqrt{q})$

Step 2.1.7.2

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

$9 p - 3 \sqrt{q p} + 3 \sqrt{p q} - ({\sqrt{q}}^{1} {\sqrt{q}}^{1})$

Step 2.1.7.3

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

$9 p - 3 \sqrt{q p} + 3 \sqrt{p q} - {\sqrt{q}}^{1 + 1}$

Step 2.1.7.4

Add $1$ and $1$ .

$9 p - 3 \sqrt{q p} + 3 \sqrt{p q} - {\sqrt{q}}^{2}$

Step 2.1.8

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

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Step 2.1.8.1

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

$9 p - 3 \sqrt{q p} + 3 \sqrt{p q} - {(q^{\frac{1}{2}})}^{2}$

Step 2.1.8.2

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

$9 p - 3 \sqrt{q p} + 3 \sqrt{p q} - q^{\frac{1}{2} \cdot 2}$

Step 2.1.8.3

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

$9 p - 3 \sqrt{q p} + 3 \sqrt{p q} - q^{\frac{2}{2}}$

Step 2.1.8.4

Cancel the common factor of $2$ .

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Step 2.1.8.4.1

Cancel the common factor.

$9 p - 3 \sqrt{q p} + 3 \sqrt{p q} - q^{\frac{2}{2}}$

Step 2.1.8.4.2

Rewrite the expression.

$9 p - 3 \sqrt{q p} + 3 \sqrt{p q} - q^{1}$

Step 2.1.8.5

Simplify.

$9 p - 3 \sqrt{q p} + 3 \sqrt{p q} - q$

Step 2.2

Add $- 3 \sqrt{q p}$ and $3 \sqrt{p q}$ .

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Step 2.2.1

Reorder $q$ and $p$ .

$9 p - 3 \sqrt{p q} + 3 \sqrt{p q} - q$

Step 2.2.2

Add $- 3 \sqrt{p q}$ and $3 \sqrt{p q}$ .

$9 p + 0 - q$

Step 2.3

Add $9 p$ and $0$ .

$9 p - q$