Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 1.1.1

Move $x$ .

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

Step 1.1.2

Multiply $x$ by $x^{5}$ .

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.3

Add $1$ and $5$ .

$\frac{x^{6} \cdot - 3 {(x^{2} + 1)}^{- 4} \cdot 2 - {(x^{2} + 1)}^{- 3} \cdot (5 x^{4})}{{(x^{5})}^{2}}$

Step 1.2

Move $- 3$ to the left of $x^{6}$ .

$\frac{- 3 \cdot x^{6} {(x^{2} + 1)}^{- 4} \cdot 2 - {(x^{2} + 1)}^{- 3} \cdot (5 x^{4})}{{(x^{5})}^{2}}$

Step 1.3

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

$\frac{- 3 x^{6} \frac{1}{{(x^{2} + 1)}^{4}} \cdot 2 - {(x^{2} + 1)}^{- 3} \cdot (5 x^{4})}{{(x^{5})}^{2}}$

Step 1.4

Multiply $- 3 x^{6} \frac{1}{{(x^{2} + 1)}^{4}}$ .

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Step 1.4.1

Combine $- 3$ and $\frac{1}{{(x^{2} + 1)}^{4}}$ .

$\frac{x^{6} \frac{- 3}{{(x^{2} + 1)}^{4}} \cdot 2 - {(x^{2} + 1)}^{- 3} \cdot (5 x^{4})}{{(x^{5})}^{2}}$

Step 1.4.2

Combine $x^{6}$ and $\frac{- 3}{{(x^{2} + 1)}^{4}}$ .

$\frac{\frac{x^{6} \cdot - 3}{{(x^{2} + 1)}^{4}} \cdot 2 - {(x^{2} + 1)}^{- 3} \cdot (5 x^{4})}{{(x^{5})}^{2}}$

Step 1.5

Move $- 3$ to the left of $x^{6}$ .

$\frac{\frac{- 3 \cdot x^{6}}{{(x^{2} + 1)}^{4}} \cdot 2 - {(x^{2} + 1)}^{- 3} \cdot (5 x^{4})}{{(x^{5})}^{2}}$

Step 1.6

Move the negative in front of the fraction.

$\frac{- \frac{3 \cdot x^{6}}{{(x^{2} + 1)}^{4}} \cdot 2 - {(x^{2} + 1)}^{- 3} \cdot (5 x^{4})}{{(x^{5})}^{2}}$

Step 1.7

Multiply $- \frac{3 x^{6}}{{(x^{2} + 1)}^{4}} \cdot 2$ .

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Step 1.7.1

Multiply $2$ by $- 1$ .

$\frac{- 2 \frac{3 x^{6}}{{(x^{2} + 1)}^{4}} - {(x^{2} + 1)}^{- 3} \cdot (5 x^{4})}{{(x^{5})}^{2}}$

Step 1.7.2

Combine $- 2$ and $\frac{3 x^{6}}{{(x^{2} + 1)}^{4}}$ .

$\frac{\frac{- 2 (3 x^{6})}{{(x^{2} + 1)}^{4}} - {(x^{2} + 1)}^{- 3} \cdot (5 x^{4})}{{(x^{5})}^{2}}$

Step 1.7.3

Multiply $3$ by $- 2$ .

$\frac{\frac{- 6 x^{6}}{{(x^{2} + 1)}^{4}} - {(x^{2} + 1)}^{- 3} \cdot (5 x^{4})}{{(x^{5})}^{2}}$

Step 1.8

Move the negative in front of the fraction.

$\frac{- \frac{6 x^{6}}{{(x^{2} + 1)}^{4}} - {(x^{2} + 1)}^{- 3} \cdot (5 x^{4})}{{(x^{5})}^{2}}$

Step 1.9

Rewrite using the commutative property of multiplication.

$\frac{- \frac{6 x^{6}}{{(x^{2} + 1)}^{4}} - 1 \cdot 5 {(x^{2} + 1)}^{- 3} x^{4}}{{(x^{5})}^{2}}$

Step 1.10

Multiply $- 1$ by $5$ .

$\frac{- \frac{6 x^{6}}{{(x^{2} + 1)}^{4}} - 5 {(x^{2} + 1)}^{- 3} x^{4}}{{(x^{5})}^{2}}$

Step 1.11

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

$\frac{- \frac{6 x^{6}}{{(x^{2} + 1)}^{4}} - 5 \frac{1}{{(x^{2} + 1)}^{3}} x^{4}}{{(x^{5})}^{2}}$

Step 1.12

Combine $- 5$ and $\frac{1}{{(x^{2} + 1)}^{3}}$ .

$\frac{- \frac{6 x^{6}}{{(x^{2} + 1)}^{4}} + \frac{- 5}{{(x^{2} + 1)}^{3}} x^{4}}{{(x^{5})}^{2}}$

Step 1.13

Move the negative in front of the fraction.

$\frac{- \frac{6 x^{6}}{{(x^{2} + 1)}^{4}} - \frac{5}{{(x^{2} + 1)}^{3}} x^{4}}{{(x^{5})}^{2}}$

Step 1.14

Combine $x^{4}$ and $\frac{5}{{(x^{2} + 1)}^{3}}$ .

$\frac{- \frac{6 x^{6}}{{(x^{2} + 1)}^{4}} - \frac{x^{4} \cdot 5}{{(x^{2} + 1)}^{3}}}{{(x^{5})}^{2}}$

Step 1.15

Move $5$ to the left of $x^{4}$ .

$\frac{- \frac{6 x^{6}}{{(x^{2} + 1)}^{4}} - \frac{5 \cdot x^{4}}{{(x^{2} + 1)}^{3}}}{{(x^{5})}^{2}}$

Step 1.16

To write $- \frac{5 x^{4}}{{(x^{2} + 1)}^{3}}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + 1}{x^{2} + 1}$ .

$\frac{- \frac{6 x^{6}}{{(x^{2} + 1)}^{4}} - \frac{5 x^{4}}{{(x^{2} + 1)}^{3}} \cdot \frac{x^{2} + 1}{x^{2} + 1}}{{(x^{5})}^{2}}$

Step 1.17

Write each expression with a common denominator of ${(x^{2} + 1)}^{4}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.17.1

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

$\frac{- \frac{6 x^{6}}{{(x^{2} + 1)}^{4}} - \frac{5 x^{4} (x^{2} + 1)}{{(x^{2} + 1)}^{3} (x^{2} + 1)}}{{(x^{5})}^{2}}$

Step 1.17.2

Multiply ${(x^{2} + 1)}^{3}$ by $x^{2} + 1$ by adding the exponents.

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Step 1.17.2.1

Multiply ${(x^{2} + 1)}^{3}$ by $x^{2} + 1$ .

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Step 1.17.2.1.1

Raise $x^{2} + 1$ to the power of $1$ .

$\frac{- \frac{6 x^{6}}{{(x^{2} + 1)}^{4}} - \frac{5 x^{4} (x^{2} + 1)}{{(x^{2} + 1)}^{3} {(x^{2} + 1)}^{1}}}{{(x^{5})}^{2}}$

Step 1.17.2.1.2

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

$\frac{- \frac{6 x^{6}}{{(x^{2} + 1)}^{4}} - \frac{5 x^{4} (x^{2} + 1)}{{(x^{2} + 1)}^{3 + 1}}}{{(x^{5})}^{2}}$

Step 1.17.2.2

Add $3$ and $1$ .

$\frac{- \frac{6 x^{6}}{{(x^{2} + 1)}^{4}} - \frac{5 x^{4} (x^{2} + 1)}{{(x^{2} + 1)}^{4}}}{{(x^{5})}^{2}}$

Step 1.18

Combine the numerators over the common denominator.

$\frac{\frac{- 6 x^{6} - 5 x^{4} (x^{2} + 1)}{{(x^{2} + 1)}^{4}}}{{(x^{5})}^{2}}$

Step 1.19

Rewrite $\frac{- 6 x^{6} - 5 x^{4} (x^{2} + 1)}{{(x^{2} + 1)}^{4}}$ in a factored form.

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Step 1.19.1

Factor $x^{4}$ out of $- 6 x^{6} - 5 x^{4} (x^{2} + 1)$ .

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Step 1.19.1.1

Factor $x^{4}$ out of $- 6 x^{6}$ .

$\frac{\frac{x^{4} (- 6 x^{2}) - 5 x^{4} (x^{2} + 1)}{{(x^{2} + 1)}^{4}}}{{(x^{5})}^{2}}$

Step 1.19.1.2

Factor $x^{4}$ out of $- 5 x^{4} (x^{2} + 1)$ .

$\frac{\frac{x^{4} (- 6 x^{2}) + x^{4} (- 5 (x^{2} + 1))}{{(x^{2} + 1)}^{4}}}{{(x^{5})}^{2}}$

Step 1.19.1.3

Factor $x^{4}$ out of $x^{4} (- 6 x^{2}) + x^{4} (- 5 (x^{2} + 1))$ .

$\frac{\frac{x^{4} (- 6 x^{2} - 5 (x^{2} + 1))}{{(x^{2} + 1)}^{4}}}{{(x^{5})}^{2}}$

Step 1.19.2

Apply the distributive property.

$\frac{\frac{x^{4} (- 6 x^{2} - 5 x^{2} - 5 \cdot 1)}{{(x^{2} + 1)}^{4}}}{{(x^{5})}^{2}}$

Step 1.19.3

Multiply $- 5$ by $1$ .

$\frac{\frac{x^{4} (- 6 x^{2} - 5 x^{2} - 5)}{{(x^{2} + 1)}^{4}}}{{(x^{5})}^{2}}$

Step 1.19.4

Subtract $5 x^{2}$ from $- 6 x^{2}$ .

$\frac{\frac{x^{4} (- 11 x^{2} - 5)}{{(x^{2} + 1)}^{4}}}{{(x^{5})}^{2}}$

Step 2

Multiply the exponents in ${(x^{5})}^{2}$ .

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Step 2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{x^{4} (- 11 x^{2} - 5)}{{(x^{2} + 1)}^{4}}}{x^{5 \cdot 2}}$

Step 2.2

Multiply $5$ by $2$ .

$\frac{\frac{x^{4} (- 11 x^{2} - 5)}{{(x^{2} + 1)}^{4}}}{x^{10}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{4} (- 11 x^{2} - 5)}{{(x^{2} + 1)}^{4}} \cdot \frac{1}{x^{10}}$

Step 4

Combine.

$\frac{x^{4} (- 11 x^{2} - 5) \cdot 1}{{(x^{2} + 1)}^{4} x^{10}}$

Step 5

Cancel the common factor of $x^{4}$ and $x^{10}$ .

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Step 5.1

Factor $x^{4}$ out of $x^{4} (- 11 x^{2} - 5) \cdot 1$ .

$\frac{x^{4} ((- 11 x^{2} - 5) \cdot 1)}{{(x^{2} + 1)}^{4} x^{10}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $x^{4}$ out of ${(x^{2} + 1)}^{4} x^{10}$ .

$\frac{x^{4} ((- 11 x^{2} - 5) \cdot 1)}{x^{4} ({(x^{2} + 1)}^{4} x^{6})}$

Step 5.2.2

Cancel the common factor.

$\frac{x^{4} ((- 11 x^{2} - 5) \cdot 1)}{x^{4} ({(x^{2} + 1)}^{4} x^{6})}$

Step 5.2.3

Rewrite the expression.

$\frac{(- 11 x^{2} - 5) \cdot 1}{{(x^{2} + 1)}^{4} x^{6}}$

Step 6

Multiply $- 11 x^{2} - 5$ by $1$ .

$\frac{- 11 x^{2} - 5}{{(x^{2} + 1)}^{4} x^{6}}$

Step 7

Factor $- 1$ out of $- 11 x^{2}$ .

$\frac{- (11 x^{2}) - 5}{{(x^{2} + 1)}^{4} x^{6}}$

Step 8

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{- (11 x^{2}) - 1 (5)}{{(x^{2} + 1)}^{4} x^{6}}$

Step 9

Factor $- 1$ out of $- (11 x^{2}) - 1 (5)$ .

$\frac{- (11 x^{2} + 5)}{{(x^{2} + 1)}^{4} x^{6}}$

Step 10

Simplify the expression.

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Step 10.1

Rewrite $- (11 x^{2} + 5)$ as $- 1 (11 x^{2} + 5)$ .

$\frac{- 1 (11 x^{2} + 5)}{{(x^{2} + 1)}^{4} x^{6}}$

Step 10.2

Move the negative in front of the fraction.

$- \frac{11 x^{2} + 5}{{(x^{2} + 1)}^{4} x^{6}}$

Step 10.3

Reorder factors in $- \frac{11 x^{2} + 5}{{(x^{2} + 1)}^{4} x^{6}}$ .

$- \frac{11 x^{2} + 5}{x^{6} {(x^{2} + 1)}^{4}}$