Precalculus Examples

$a^{\frac{1}{2}} (a^{\frac{3}{2}} - 2 a^{\frac{1}{2}})$

Step 1

Apply the distributive property.

$a^{\frac{1}{2}} a^{\frac{3}{2}} + a^{\frac{1}{2}} (- 2 a^{\frac{1}{2}})$

Step 2

Multiply $a^{\frac{1}{2}}$ by $a^{\frac{3}{2}}$ by adding the exponents.

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Step 2.1

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

$a^{\frac{1}{2} + \frac{3}{2}} + a^{\frac{1}{2}} (- 2 a^{\frac{1}{2}})$

Step 2.2

Combine the numerators over the common denominator.

$a^{\frac{1 + 3}{2}} + a^{\frac{1}{2}} (- 2 a^{\frac{1}{2}})$

Step 2.3

Add $1$ and $3$ .

$a^{\frac{4}{2}} + a^{\frac{1}{2}} (- 2 a^{\frac{1}{2}})$

Step 2.4

Divide $4$ by $2$ .

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

Step 3

Rewrite using the commutative property of multiplication.

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

Step 4

Simplify each term.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

Combine the numerators over the common denominator.

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

Step 4.1.4

Add $1$ and $1$ .

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

Step 4.1.5

Divide $2$ by $2$ .

$a^{2} - 2 a^{1}$

Step 4.2

Simplify $- 2 a^{1}$ .

$a^{2} - 2 a$