Pre-Algebra Examples

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

Step 1

Expand $(q + q^{\frac{1}{2}}) (q - q^{\frac{1}{2}})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

$q \cdot q + q (- q^{\frac{1}{2}}) + q^{\frac{1}{2}} (q - q^{\frac{1}{2}})$

Step 1.3

Apply the distributive property.

$q \cdot q + q (- q^{\frac{1}{2}}) + q^{\frac{1}{2}} q + q^{\frac{1}{2}} (- q^{\frac{1}{2}})$

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $q \cdot q + q (- q^{\frac{1}{2}}) + q^{\frac{1}{2}} q + q^{\frac{1}{2}} (- q^{\frac{1}{2}})$ .

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Step 2.1.1

Reorder the factors in the terms $q (- q^{\frac{1}{2}})$ and $q^{\frac{1}{2}} q$ .

$q \cdot q - q^{\frac{1}{2}} q + q^{\frac{1}{2}} q + q^{\frac{1}{2}} (- q^{\frac{1}{2}})$

Step 2.1.2

Add $- q^{\frac{1}{2}} q$ and $q^{\frac{1}{2}} q$ .

$q \cdot q + 0 + q^{\frac{1}{2}} (- q^{\frac{1}{2}})$

Step 2.1.3

Add $q \cdot q$ and $0$ .

$q \cdot q + q^{\frac{1}{2}} (- q^{\frac{1}{2}})$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $q$ by $q$ .

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

Step 2.2.2

Rewrite using the commutative property of multiplication.

$q^{2} - q^{\frac{1}{2}} q^{\frac{1}{2}}$

Step 2.2.3

Multiply $q^{\frac{1}{2}}$ by $q^{\frac{1}{2}}$ by adding the exponents.

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Step 2.2.3.1

Move $q^{\frac{1}{2}}$ .

$q^{2} - (q^{\frac{1}{2}} q^{\frac{1}{2}})$

Step 2.2.3.2

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

$q^{2} - q^{\frac{1}{2} + \frac{1}{2}}$

Step 2.2.3.3

Combine the numerators over the common denominator.

$q^{2} - q^{\frac{1 + 1}{2}}$

Step 2.2.3.4

Add $1$ and $1$ .

$q^{2} - q^{\frac{2}{2}}$

Step 2.2.3.5

Divide $2$ by $2$ .

$q^{2} - q^{1}$

Step 2.2.4

Simplify $- q^{1}$ .

$q^{2} - q$