Precalculus Examples

$\frac{\frac{3 x^{\frac{3}{2}}}{2} - 6 x^{\frac{3}{2}}}{9 x^{4}} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 1

To write $- 6 x^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{3 x^{\frac{3}{2}}}{2} - 6 x^{\frac{3}{2}} \cdot \frac{2}{2}}{9 x^{4}} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 2

Simplify terms.

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Step 2.1

Combine $- 6 x^{\frac{3}{2}}$ and $\frac{2}{2}$ .

$\frac{\frac{3 x^{\frac{3}{2}}}{2} + \frac{- 6 x^{\frac{3}{2}} \cdot 2}{2}}{9 x^{4}} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{\frac{3 x^{\frac{3}{2}} - 6 x^{\frac{3}{2}} \cdot 2}{2}}{9 x^{4}} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3

Simplify each term.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 x^{\frac{3}{2}} - 6 x^{\frac{3}{2}} \cdot 2}{2} \cdot \frac{1}{9 x^{4}} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

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

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Step 3.2.1.1

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

$\frac{3 x^{\frac{3}{2}} - 6 \cdot 2 x^{\frac{3}{2}}}{2} \cdot \frac{1}{9 x^{4}} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.2.1.2

Factor $3 x^{\frac{3}{2}}$ out of $3 x^{\frac{3}{2}}$ .

$\frac{3 x^{\frac{3}{2}} (1) - 6 \cdot 2 x^{\frac{3}{2}}}{2} \cdot \frac{1}{9 x^{4}} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.2.1.3

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

$\frac{3 x^{\frac{3}{2}} (1) + 3 x^{\frac{3}{2}} (- 2 \cdot 2)}{2} \cdot \frac{1}{9 x^{4}} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.2.1.4

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

$\frac{3 x^{\frac{3}{2}} (1 - 2 \cdot 2)}{2} \cdot \frac{1}{9 x^{4}} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.2.2

Multiply $- 2$ by $2$ .

$\frac{3 x^{\frac{3}{2}} (1 - 4)}{2} \cdot \frac{1}{9 x^{4}} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.2.3

Subtract $4$ from $1$ .

$\frac{3 x^{\frac{3}{2}} \cdot - 3}{2} \cdot \frac{1}{9 x^{4}} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.2.4

Multiply $- 3$ by $3$ .

$\frac{- 9 x^{\frac{3}{2}}}{2} \cdot \frac{1}{9 x^{4}} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.3

Combine.

$\frac{- 9 x^{\frac{3}{2}} \cdot 1}{2 (9 x^{4})} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.4

Move $x^{\frac{3}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{- 9}{2 (9 x^{4}) x^{- \frac{3}{2}}} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.5

Multiply $x^{4}$ by $x^{- \frac{3}{2}}$ by adding the exponents.

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Step 3.5.1

Move $x^{- \frac{3}{2}}$ .

$\frac{- 9}{2 (9 (x^{- \frac{3}{2}} x^{4}))} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.5.2

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

$\frac{- 9}{2 (9 x^{- \frac{3}{2} + 4})} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.5.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- 9}{2 (9 x^{- \frac{3}{2} + 4 \cdot \frac{2}{2}})} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.5.4

Combine $4$ and $\frac{2}{2}$ .

$\frac{- 9}{2 (9 x^{- \frac{3}{2} + \frac{4 \cdot 2}{2}})} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.5.5

Combine the numerators over the common denominator.

$\frac{- 9}{2 (9 x^{\frac{- 3 + 4 \cdot 2}{2}})} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.5.6

Simplify the numerator.

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Step 3.5.6.1

Multiply $4$ by $2$ .

$\frac{- 9}{2 (9 x^{\frac{- 3 + 8}{2}})} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.5.6.2

Add $- 3$ and $8$ .

$\frac{- 9}{2 (9 x^{\frac{5}{2}})} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.6

Factor $9$ out of $- 9$ .

$\frac{9 \cdot - 1}{2 (9 x^{\frac{5}{2}})} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.7

Cancel the common factors.

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Step 3.7.1

Factor $9$ out of $2 (9 x^{\frac{5}{2}})$ .

$\frac{9 \cdot - 1}{9 (2 (x^{\frac{5}{2}}))} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.7.2

Cancel the common factor.

$\frac{9 \cdot - 1}{9 (2 (x^{\frac{5}{2}}))} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.7.3

Rewrite the expression.

$\frac{- 1}{2 (x^{\frac{5}{2}})} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.8

Move the negative in front of the fraction.

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Factor $3 x^{\frac{1}{2}}$ out of $6 x^{\frac{1}{2}} - 9 x^{\frac{1}{2}} \ln (x)$ .

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Step 3.9.1.1

Factor $3 x^{\frac{1}{2}}$ out of $6 x^{\frac{1}{2}}$ .

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 x^{\frac{1}{2}} (2) - 9 x^{\frac{1}{2}} \ln (x)}{36 x^{3}}$

Step 3.9.1.2

Factor $3 x^{\frac{1}{2}}$ out of $- 9 x^{\frac{1}{2}} \ln (x)$ .

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 x^{\frac{1}{2}} (2) + 3 x^{\frac{1}{2}} (- 3 \ln (x))}{36 x^{3}}$

Step 3.9.1.3

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

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 x^{\frac{1}{2}} (2 - 3 \ln (x))}{36 x^{3}}$

Step 3.9.2

Simplify $- 3 \ln (x)$ by moving $3$ inside the logarithm.

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 x^{\frac{1}{2}} (2 - \ln (x^{3}))}{36 x^{3}}$

Step 3.10

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 (2 - \ln (x^{3}))}{36 x^{3} x^{- \frac{1}{2}}}$

Step 3.11

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

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Step 3.11.1

Move $x^{- \frac{1}{2}}$ .

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 (2 - \ln (x^{3}))}{36 (x^{- \frac{1}{2}} x^{3})}$

Step 3.11.2

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

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 (2 - \ln (x^{3}))}{36 x^{- \frac{1}{2} + 3}}$

Step 3.11.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 (2 - \ln (x^{3}))}{36 x^{- \frac{1}{2} + 3 \cdot \frac{2}{2}}}$

Step 3.11.4

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

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 (2 - \ln (x^{3}))}{36 x^{- \frac{1}{2} + \frac{3 \cdot 2}{2}}}$

Step 3.11.5

Combine the numerators over the common denominator.

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 (2 - \ln (x^{3}))}{36 x^{\frac{- 1 + 3 \cdot 2}{2}}}$

Step 3.11.6

Simplify the numerator.

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Step 3.11.6.1

Multiply $3$ by $2$ .

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 (2 - \ln (x^{3}))}{36 x^{\frac{- 1 + 6}{2}}}$

Step 3.11.6.2

Add $- 1$ and $6$ .

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 (2 - \ln (x^{3}))}{36 x^{\frac{5}{2}}}$

Step 3.12

Expand $\ln (x^{3})$ by moving $3$ outside the logarithm.

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 (2 - (3 \ln (x)))}{36 x^{\frac{5}{2}}}$

Step 3.13

Cancel the common factors.

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Step 3.13.1

Factor $3$ out of $36 x^{\frac{5}{2}}$ .

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 (2 - (3 \ln (x)))}{3 (12 x^{\frac{5}{2}})}$

Step 3.13.2

Cancel the common factor.

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{3 (2 - (3 \ln (x)))}{3 (12 x^{\frac{5}{2}})}$

Step 3.13.3

Rewrite the expression.

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{2 - (3 \ln (x))}{12 x^{\frac{5}{2}}}$

Step 3.14

Simplify the numerator.

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Step 3.14.1

Multiply $- 1$ by $3$ .

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{2 - 3 \ln (x)}{12 x^{\frac{5}{2}}}$

Step 3.14.2

Simplify $- 3 \ln (x)$ by moving $3$ inside the logarithm.

$- \frac{1}{2 x^{\frac{5}{2}}} - \frac{2 - \ln (x^{3})}{12 x^{\frac{5}{2}}}$

Step 4

To write $- \frac{1}{2 x^{\frac{5}{2}}}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{1}{2 x^{\frac{5}{2}}} \cdot \frac{6}{6} - \frac{2 - \ln (x^{3})}{12 x^{\frac{5}{2}}}$

Step 5

Write each expression with a common denominator of $12 x^{\frac{5}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{1}{2 x^{\frac{5}{2}}}$ by $\frac{6}{6}$ .

$- \frac{6}{2 x^{\frac{5}{2}} \cdot 6} - \frac{2 - \ln (x^{3})}{12 x^{\frac{5}{2}}}$

Step 5.2

Multiply $6$ by $2$ .

$- \frac{6}{12 x^{\frac{5}{2}}} - \frac{2 - \ln (x^{3})}{12 x^{\frac{5}{2}}}$

Step 6

Combine the numerators over the common denominator.

$\frac{- 6 - (2 - \ln (x^{3}))}{12 x^{\frac{5}{2}}}$

Step 7

Simplify the numerator.

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Step 7.1

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

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Step 7.1.1

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{- 1 (6) - (2 - \ln (x^{3}))}{12 x^{\frac{5}{2}}}$

Step 7.1.2

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

$\frac{- 1 (6 + 2 - \ln (x^{3}))}{12 x^{\frac{5}{2}}}$

Step 7.1.3

Rewrite $- 1 (6 + 2 - \ln (x^{3}))$ as $- (6 + 2 - \ln (x^{3}))$ .

$\frac{- (6 + 2 - \ln (x^{3}))}{12 x^{\frac{5}{2}}}$

Step 7.2

Add $6$ and $2$ .

$\frac{- (8 - \ln (x^{3}))}{12 x^{\frac{5}{2}}}$

Step 8

Move the negative in front of the fraction.

$- \frac{8 - \ln (x^{3})}{12 x^{\frac{5}{2}}}$