Precalculus Examples

$\frac{1}{10} \cdot {(2 x + 1)}^{\frac{5}{2}} - \frac{1}{6} \cdot {(2 x + 1)}^{\frac{3}{2}}$

Step 1

Factor ${(2 x + 1)}^{\frac{3}{2}}$ out of $\frac{1}{10} \cdot {(2 x + 1)}^{\frac{5}{2}} - \frac{1}{6} \cdot {(2 x + 1)}^{\frac{3}{2}}$ .

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Step 1.1

Factor ${(2 x + 1)}^{\frac{3}{2}}$ out of $\frac{1}{10} \cdot {(2 x + 1)}^{\frac{5}{2}}$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{1}{10} \cdot {(2 x + 1)}^{\frac{2}{2}}) - \frac{1}{6} \cdot {(2 x + 1)}^{\frac{3}{2}}$

Step 1.2

Factor ${(2 x + 1)}^{\frac{3}{2}}$ out of $- \frac{1}{6} \cdot {(2 x + 1)}^{\frac{3}{2}}$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{1}{10} \cdot {(2 x + 1)}^{\frac{2}{2}}) + {(2 x + 1)}^{\frac{3}{2}} \cdot (- \frac{1}{6})$

Step 1.3

Factor ${(2 x + 1)}^{\frac{3}{2}}$ out of ${(2 x + 1)}^{\frac{3}{2}} (\frac{1}{10} \cdot {(2 x + 1)}^{\frac{2}{2}}) + {(2 x + 1)}^{\frac{3}{2}} \cdot (- \frac{1}{6})$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{1}{10} \cdot {(2 x + 1)}^{\frac{2}{2}} - \frac{1}{6})$

Step 2

Divide $2$ by $2$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{1}{10} \cdot {(2 x + 1)}^{1} - \frac{1}{6})$

Step 3

Simplify.

${(2 x + 1)}^{\frac{3}{2}} (\frac{1}{10} \cdot (2 x + 1) - \frac{1}{6})$

Step 4

Apply the distributive property.

${(2 x + 1)}^{\frac{3}{2}} (\frac{1}{10} (2 x) + \frac{1}{10} \cdot 1 - \frac{1}{6})$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Factor $2$ out of $10$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{1}{2 (5)} (2 x) + \frac{1}{10} \cdot 1 - \frac{1}{6})$

Step 5.2

Factor $2$ out of $2 x$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{1}{2 (5)} (2 (x)) + \frac{1}{10} \cdot 1 - \frac{1}{6})$

Step 5.3

Cancel the common factor.

${(2 x + 1)}^{\frac{3}{2}} (\frac{1}{2 \cdot 5} (2 x) + \frac{1}{10} \cdot 1 - \frac{1}{6})$

Step 5.4

Rewrite the expression.

${(2 x + 1)}^{\frac{3}{2}} (\frac{1}{5} x + \frac{1}{10} \cdot 1 - \frac{1}{6})$

Step 6

Combine $\frac{1}{5}$ and $x$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{x}{5} + \frac{1}{10} \cdot 1 - \frac{1}{6})$

Step 7

Multiply $\frac{1}{10}$ by $1$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{x}{5} + \frac{1}{10} - \frac{1}{6})$

Step 8

To write $\frac{1}{10}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{x}{5} + \frac{1}{10} \cdot \frac{3}{3} - \frac{1}{6})$

Step 9

To write $- \frac{1}{6}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{x}{5} + \frac{1}{10} \cdot \frac{3}{3} - \frac{1}{6} \cdot \frac{5}{5})$

Step 10

Write each expression with a common denominator of $30$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1

Multiply $\frac{1}{10}$ by $\frac{3}{3}$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{x}{5} + \frac{3}{10 \cdot 3} - \frac{1}{6} \cdot \frac{5}{5})$

Step 10.2

Multiply $10$ by $3$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{x}{5} + \frac{3}{30} - \frac{1}{6} \cdot \frac{5}{5})$

Step 10.3

Multiply $\frac{1}{6}$ by $\frac{5}{5}$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{x}{5} + \frac{3}{30} - \frac{5}{6 \cdot 5})$

Step 10.4

Multiply $6$ by $5$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{x}{5} + \frac{3}{30} - \frac{5}{30})$

Step 11

Combine the numerators over the common denominator.

${(2 x + 1)}^{\frac{3}{2}} (\frac{x}{5} + \frac{3 - 5}{30})$

Step 12

Subtract $5$ from $3$ .

${(2 x + 1)}^{\frac{3}{2}} (\frac{x}{5} + \frac{- 2}{30})$