Precalculus Examples

$4 x^{2} (- \frac{1}{3} \cdot ({(9 - x^{2})}^{- \frac{5}{2}} (- 3 x))) + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1

Move $x$ .

$4 (x \cdot x^{2}) (- \frac{1}{3} \cdot ({(9 - x^{2})}^{- \frac{5}{2}} \cdot (- 3))) + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 1.2

Multiply $x$ by $x^{2}$ .

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Step 1.2.1

Raise $x$ to the power of $1$ .

$4 (x^{1} x^{2}) (- \frac{1}{3} \cdot ({(9 - x^{2})}^{- \frac{5}{2}} \cdot (- 3))) + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 1.2.2

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

$4 x^{1 + 2} (- \frac{1}{3} \cdot ({(9 - x^{2})}^{- \frac{5}{2}} \cdot (- 3))) + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 1.3

Add $1$ and $2$ .

$4 x^{3} (- \frac{1}{3} \cdot ({(9 - x^{2})}^{- \frac{5}{2}} \cdot (- 3))) + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 2

Cancel the common factor of $3$ .

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Step 2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$4 x^{3} (\frac{- 1}{3} \cdot ({(9 - x^{2})}^{- \frac{5}{2}} \cdot (- 3))) + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 2.2

Factor $3$ out of ${(9 - x^{2})}^{- \frac{5}{2}} \cdot (- 3)$ .

$4 x^{3} (\frac{- 1}{3} \cdot (3 ({(9 - x^{2})}^{- \frac{5}{2}} \cdot (- 1)))) + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 2.3

Cancel the common factor.

$4 x^{3} (\frac{- 1}{3} \cdot (3 ({(9 - x^{2})}^{- \frac{5}{2}} \cdot (- 1)))) + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 2.4

Rewrite the expression.

$4 x^{3} (- 1 \cdot ({(9 - x^{2})}^{- \frac{5}{2}} \cdot (- 1))) + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 3

Multiply $- 1$ by $- 1$ .

$4 x^{3} (1 \cdot {(9 - x^{2})}^{- \frac{5}{2}}) + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 4

Multiply ${(9 - x^{2})}^{- \frac{5}{2}}$ by $1$ .

$4 x^{3} {(9 - x^{2})}^{- \frac{5}{2}} + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 x^{3} \frac{1}{{(9 - x^{2})}^{\frac{5}{2}}} + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 6

Multiply $4 x^{3} \frac{1}{{(9 - x^{2})}^{\frac{5}{2}}}$ .

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Step 6.1

Combine $4$ and $\frac{1}{{(9 - x^{2})}^{\frac{5}{2}}}$ .

$x^{3} \frac{4}{{(9 - x^{2})}^{\frac{5}{2}}} + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 6.2

Combine $x^{3}$ and $\frac{4}{{(9 - x^{2})}^{\frac{5}{2}}}$ .

$\frac{x^{3} \cdot 4}{{(9 - x^{2})}^{\frac{5}{2}}} + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 7

Move $4$ to the left of $x^{3}$ .

$\frac{4 \cdot x^{3}}{{(9 - x^{2})}^{\frac{5}{2}}} + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 8

Multiply $4$ by $x^{3}$ .

$\frac{4 x^{3}}{{(9 - x^{2})}^{\frac{5}{2}}} + {(9 - x^{2})}^{- \frac{3}{2}} (8 x)$

Step 9

Rewrite using the commutative property of multiplication.

$\frac{4 x^{3}}{{(9 - x^{2})}^{\frac{5}{2}}} + 8 {(9 - x^{2})}^{- \frac{3}{2}} x$

Step 10

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4 x^{3}}{{(9 - x^{2})}^{\frac{5}{2}}} + 8 \frac{1}{{(9 - x^{2})}^{\frac{3}{2}}} x$

Step 11

Combine $8$ and $\frac{1}{{(9 - x^{2})}^{\frac{3}{2}}}$ .

$\frac{4 x^{3}}{{(9 - x^{2})}^{\frac{5}{2}}} + \frac{8}{{(9 - x^{2})}^{\frac{3}{2}}} x$

Step 12

Combine $\frac{8}{{(9 - x^{2})}^{\frac{3}{2}}}$ and $x$ .

$\frac{4 x^{3}}{{(9 - x^{2})}^{\frac{5}{2}}} + \frac{8 x}{{(9 - x^{2})}^{\frac{3}{2}}}$