Algebra Examples

$\frac{3}{2} u^{2} v^{4} - 2 \frac{1}{2} u^{3} v^{5} + \frac{1}{2} u^{4} v^{2}$

Step 1

Remove parentheses.

$\frac{3}{2} \cdot u^{2} v^{4} - 2 \frac{1}{2} u^{3} v^{5} + \frac{1}{2} \cdot (u^{4} v^{2})$

Step 2

Remove parentheses.

$\frac{3}{2} \cdot u^{2} v^{4} - 2 \frac{1}{2} u^{3} v^{5} + \frac{1}{2} \cdot u^{4} v^{2}$

Step 3

Factor $- u^{2} v^{2}$ out of $\frac{3}{2} \cdot u^{2} v^{4} - 2 \frac{1}{2} u^{3} v^{5} + \frac{1}{2} \cdot u^{4} v^{2}$ .

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Step 3.1

Reorder the expression.

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Step 3.1.1

Move $\frac{3}{2}$ .

$u^{2} v^{4} \frac{3}{2} - 2 \frac{1}{2} u^{3} v^{5} + \frac{1}{2} \cdot u^{4} v^{2}$

Step 3.1.2

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

$u^{2} v^{4} \frac{3}{2} - 1 u^{3} v^{5} (2 \frac{1}{2}) + \frac{1}{2} \cdot u^{4} v^{2}$

Step 3.1.3

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

$u^{2} v^{4} \frac{3}{2} - 1 u^{3} v^{5} (2 \frac{1}{2}) + u^{4} v^{2} \frac{1}{2}$

Step 3.1.4

Move $u^{2} v^{4} \frac{3}{2}$ .

$- 1 u^{3} v^{5} (2 \frac{1}{2}) + u^{4} v^{2} \frac{1}{2} + u^{2} v^{4} \frac{3}{2}$

Step 3.2

Factor $- u^{2} v^{2}$ out of $- 1 u^{3} v^{5} (2 \frac{1}{2})$ .

$- u^{2} v^{2} (u v^{3} (2 \frac{1}{2})) + u^{4} v^{2} \frac{1}{2} + u^{2} v^{4} \frac{3}{2}$

Step 3.3

Factor $- u^{2} v^{2}$ out of $u^{4} v^{2} \frac{1}{2}$ .

$- u^{2} v^{2} (u v^{3} (2 \frac{1}{2})) - u^{2} v^{2} (- u^{2} \frac{1}{2}) + u^{2} v^{4} \frac{3}{2}$

Step 3.4

Factor $- u^{2} v^{2}$ out of $u^{2} v^{4} \frac{3}{2}$ .

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

Step 3.5

Factor $- u^{2} v^{2}$ out of $- u^{2} v^{2} (u v^{3} (2 \frac{1}{2})) - u^{2} v^{2} (- u^{2} \frac{1}{2})$ .

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

Step 3.6

Factor $- u^{2} v^{2}$ out of $- u^{2} v^{2} (u v^{3} (2 \frac{1}{2}) - u^{2} \frac{1}{2}) - u^{2} v^{2} (- 1 v^{2} \frac{3}{2})$ .

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

Step 4

Combine exponents.

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Step 4.1

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

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

Step 4.2

Combine $\frac{3}{2}$ and $v^{2}$ .

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

Step 5

Convert $2 \frac{1}{2}$ to an improper fraction.

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Step 5.1

A mixed number is an addition of its whole and fractional parts.

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

Step 5.2

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

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Step 5.2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- u^{2} v^{2} (u v^{3} (2 \cdot \frac{2}{2} + \frac{1}{2}) - \frac{u^{2}}{2} - 1 \frac{3 v^{2}}{2})$

Step 5.2.2

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

$- u^{2} v^{2} (u v^{3} (\frac{2 \cdot 2}{2} + \frac{1}{2}) - \frac{u^{2}}{2} - 1 \frac{3 v^{2}}{2})$

Step 5.2.3

Combine the numerators over the common denominator.

$- u^{2} v^{2} (u v^{3} \frac{2 \cdot 2 + 1}{2} - \frac{u^{2}}{2} - 1 \frac{3 v^{2}}{2})$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

Multiply $2$ by $2$ .

$- u^{2} v^{2} (u v^{3} \frac{4 + 1}{2} - \frac{u^{2}}{2} - 1 \frac{3 v^{2}}{2})$

Step 5.2.4.2

Add $4$ and $1$ .

$- u^{2} v^{2} (u v^{3} \frac{5}{2} - \frac{u^{2}}{2} - 1 \frac{3 v^{2}}{2})$

Step 6

Multiply $u v^{3} \frac{5}{2}$ .

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Step 6.1

Combine $u$ and $\frac{5}{2}$ .

$- u^{2} v^{2} (v^{3} \frac{u \cdot 5}{2} - \frac{u^{2}}{2} - 1 \frac{3 v^{2}}{2})$

Step 6.2

Combine $v^{3}$ and $\frac{u \cdot 5}{2}$ .

$- u^{2} v^{2} (\frac{v^{3} (u \cdot 5)}{2} - \frac{u^{2}}{2} - 1 \frac{3 v^{2}}{2})$

Step 7

Remove parentheses.

$- u^{2} v^{2} (\frac{v^{3} u \cdot 5}{2} - \frac{u^{2}}{2} - 1 \frac{3 v^{2}}{2})$

Step 8

Move $5$ to the left of $v^{3} u$ .

$- u^{2} v^{2} (\frac{5 \cdot (v^{3} u)}{2} - \frac{u^{2}}{2} - 1 \frac{3 v^{2}}{2})$

Step 9

Remove parentheses.

$- u^{2} v^{2} (\frac{5 \cdot v^{3} u}{2} - \frac{u^{2}}{2} - 1 \frac{3 v^{2}}{2})$

Step 10

Multiply $5$ by $v^{3}$ .

$- u^{2} v^{2} (\frac{5 v^{3} u}{2} - \frac{u^{2}}{2} - 1 \frac{3 v^{2}}{2})$

Step 11

Rewrite $- 1 \frac{3 v^{2}}{2}$ as $- \frac{3 v^{2}}{2}$ .

$- u^{2} v^{2} (\frac{5 v^{3} u}{2} - \frac{u^{2}}{2} - \frac{3 v^{2}}{2})$