Calculus Examples

${(\frac{1}{x} - \sqrt{x})}^{5}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

$1 - 5 - 10 - 10 - 5 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(\frac{1}{x} - \sqrt{x})}^{5}$ , $n = 5$ so the coefficients of the expansion will correspond with line $6$ .

Step 2

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 5 - 10 - 10 - 5 - 1$ .

$1 a^{5} b^{0} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + 1 a^{0} b^{5}$

Step 3

Substitute the actual values of $a$ $\frac{1}{x}$ and $b$ $- \sqrt{x}$ into the expression.

$1 {(\frac{1}{x})}^{5} {(- \sqrt{x})}^{0} + 5 {(\frac{1}{x})}^{4} {(- \sqrt{x})}^{1} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4

Simplify each term.

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Step 4.1

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

${(\frac{1}{x})}^{5} {(- \sqrt{x})}^{0} + 5 {(\frac{1}{x})}^{4} {(- \sqrt{x})}^{1} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.2

Apply the product rule to $\frac{1}{x}$ .

$\frac{1^{5}}{x^{5}} {(- \sqrt{x})}^{0} + 5 {(\frac{1}{x})}^{4} {(- \sqrt{x})}^{1} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.3

One to any power is one.

$\frac{1}{x^{5}} {(- \sqrt{x})}^{0} + 5 {(\frac{1}{x})}^{4} {(- \sqrt{x})}^{1} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.4

Apply the product rule to $- \sqrt{x}$ .

$\frac{1}{x^{5}} ({(- 1)}^{0} {\sqrt{x}}^{0}) + 5 {(\frac{1}{x})}^{4} {(- \sqrt{x})}^{1} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.5

Rewrite using the commutative property of multiplication.

${(- 1)}^{0} \frac{1}{x^{5}} {\sqrt{x}}^{0} + 5 {(\frac{1}{x})}^{4} {(- \sqrt{x})}^{1} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.6

Anything raised to $0$ is $1$ .

$1 \frac{1}{x^{5}} {\sqrt{x}}^{0} + 5 {(\frac{1}{x})}^{4} {(- \sqrt{x})}^{1} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.7

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

$\frac{1}{x^{5}} {\sqrt{x}}^{0} + 5 {(\frac{1}{x})}^{4} {(- \sqrt{x})}^{1} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.8

Anything raised to $0$ is $1$ .

$\frac{1}{x^{5}} \cdot 1 + 5 {(\frac{1}{x})}^{4} {(- \sqrt{x})}^{1} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.9

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

$\frac{1}{x^{5}} + 5 {(\frac{1}{x})}^{4} {(- \sqrt{x})}^{1} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.10

Apply the product rule to $\frac{1}{x}$ .

$\frac{1}{x^{5}} + 5 \frac{1^{4}}{x^{4}} {(- \sqrt{x})}^{1} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.11

One to any power is one.

$\frac{1}{x^{5}} + 5 \frac{1}{x^{4}} {(- \sqrt{x})}^{1} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.12

Combine $5$ and $\frac{1}{x^{4}}$ .

$\frac{1}{x^{5}} + \frac{5}{x^{4}} {(- \sqrt{x})}^{1} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.13

Simplify.

$\frac{1}{x^{5}} + \frac{5}{x^{4}} (- \sqrt{x}) + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.14

Rewrite using the commutative property of multiplication.

$\frac{1}{x^{5}} - \frac{5}{x^{4}} \sqrt{x} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.15

Combine $\sqrt{x}$ and $\frac{5}{x^{4}}$ .

$\frac{1}{x^{5}} - \frac{\sqrt{x} \cdot 5}{x^{4}} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.16

Move $5$ to the left of $\sqrt{x}$ .

$\frac{1}{x^{5}} - \frac{5 \cdot \sqrt{x}}{x^{4}} + 10 {(\frac{1}{x})}^{3} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.17

Apply the product rule to $\frac{1}{x}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + 10 \frac{1^{3}}{x^{3}} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.18

One to any power is one.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + 10 \frac{1}{x^{3}} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.19

Combine $10$ and $\frac{1}{x^{3}}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{3}} {(- \sqrt{x})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.20

Apply the product rule to $- \sqrt{x}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{3}} ({(- 1)}^{2} {\sqrt{x}}^{2}) + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.21

Rewrite using the commutative property of multiplication.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + {(- 1)}^{2} \frac{10}{x^{3}} {\sqrt{x}}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.22

Raise $- 1$ to the power of $2$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + 1 \frac{10}{x^{3}} {\sqrt{x}}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.23

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

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{3}} {\sqrt{x}}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.24

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 4.24.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{3}} {(x^{\frac{1}{2}})}^{2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.24.2

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

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{3}} x^{\frac{1}{2} \cdot 2} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.24.3

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

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{3}} x^{\frac{2}{2}} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.24.4

Cancel the common factor of $2$ .

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Step 4.24.4.1

Cancel the common factor.

Step 4.24.4.2

Rewrite the expression.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{3}} x^{1} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.24.5

Simplify.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{3}} x + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.25

Cancel the common factor of $x$ .

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Step 4.25.1

Factor $x$ out of $x^{3}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x \cdot x^{2}} x + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.25.2

Cancel the common factor.

Step 4.25.3

Rewrite the expression.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} + 10 {(\frac{1}{x})}^{2} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.26

Apply the product rule to $\frac{1}{x}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} + 10 \frac{1^{2}}{x^{2}} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.27

One to any power is one.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} + 10 \frac{1}{x^{2}} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.28

Combine $10$ and $\frac{1}{x^{2}}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} + \frac{10}{x^{2}} {(- \sqrt{x})}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.29

Apply the product rule to $- \sqrt{x}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} + \frac{10}{x^{2}} ({(- 1)}^{3} {\sqrt{x}}^{3}) + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.30

Rewrite using the commutative property of multiplication.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} + {(- 1)}^{3} \frac{10}{x^{2}} {\sqrt{x}}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.31

Raise $- 1$ to the power of $3$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10}{x^{2}} {\sqrt{x}}^{3} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.32

Rewrite ${\sqrt{x}}^{3}$ as $\sqrt{x^{3}}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10}{x^{2}} \sqrt{x^{3}} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.33

Factor out $x^{2}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10}{x^{2}} \sqrt{x^{2} x} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.34

Pull terms out from under the radical.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10}{x^{2}} (x \sqrt{x}) + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.35

Cancel the common factor of $x$ .

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Step 4.35.1

Move the leading negative in $- \frac{10}{x^{2}}$ into the numerator.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} + \frac{- 10}{x^{2}} (x \sqrt{x}) + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.35.2

Factor $x$ out of $x^{2}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} + \frac{- 10}{x \cdot x} (x \sqrt{x}) + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.35.3

Factor $x$ out of $x \sqrt{x}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} + \frac{- 10}{x \cdot x} (x (\sqrt{x})) + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.35.4

Cancel the common factor.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} + \frac{- 10}{x \cdot x} (x \sqrt{x}) + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.35.5

Rewrite the expression.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} + \frac{- 10}{x} \sqrt{x} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.36

Combine $\frac{- 10}{x}$ and $\sqrt{x}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} + \frac{- 10 \sqrt{x}}{x} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.37

Move the negative in front of the fraction.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{(10) \sqrt{x}}{x} + 5 {(\frac{1}{x})}^{1} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.38

Simplify.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + 5 \frac{1}{x} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.39

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

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + \frac{5}{x} {(- \sqrt{x})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.40

Apply the product rule to $- \sqrt{x}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + \frac{5}{x} ({(- 1)}^{4} {\sqrt{x}}^{4}) + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.41

Rewrite using the commutative property of multiplication.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + {(- 1)}^{4} \frac{5}{x} {\sqrt{x}}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.42

Raise $- 1$ to the power of $4$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + 1 \frac{5}{x} {\sqrt{x}}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.43

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

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + \frac{5}{x} {\sqrt{x}}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.44

Rewrite ${\sqrt{x}}^{4}$ as $x^{2}$ .

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Step 4.44.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + \frac{5}{x} {(x^{\frac{1}{2}})}^{4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.44.2

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

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + \frac{5}{x} x^{\frac{1}{2} \cdot 4} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.44.3

Combine $\frac{1}{2}$ and $4$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + \frac{5}{x} x^{\frac{4}{2}} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.44.4

Cancel the common factor of $4$ and $2$ .

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Step 4.44.4.1

Factor $2$ out of $4$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + \frac{5}{x} x^{\frac{2 \cdot 2}{2}} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.44.4.2

Cancel the common factors.

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Step 4.44.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + \frac{5}{x} x^{\frac{2 \cdot 2}{2 (1)}} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.44.4.2.2

Cancel the common factor.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + \frac{5}{x} x^{\frac{2 \cdot 2}{2 \cdot 1}} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.44.4.2.3

Rewrite the expression.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + \frac{5}{x} x^{\frac{2}{1}} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.44.4.2.4

Divide $2$ by $1$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + \frac{5}{x} x^{2} + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.45

Cancel the common factor of $x$ .

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Step 4.45.1

Factor $x$ out of $x^{2}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + \frac{5}{x} (x \cdot x) + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.45.2

Cancel the common factor.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + \frac{5}{x} (x \cdot x) + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.45.3

Rewrite the expression.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + 5 x + 1 {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.46

Multiply ${(\frac{1}{x})}^{0}$ by $1$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + 5 x + {(\frac{1}{x})}^{0} {(- \sqrt{x})}^{5}$

Step 4.47

Apply the product rule to $\frac{1}{x}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + 5 x + \frac{1^{0}}{x^{0}} {(- \sqrt{x})}^{5}$

Step 4.48

Anything raised to $0$ is $1$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + 5 x + \frac{1}{x^{0}} {(- \sqrt{x})}^{5}$

Step 4.49

Anything raised to $0$ is $1$ .

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

Step 4.50

Divide $1$ by $1$ .

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

Step 4.51

Multiply ${(- \sqrt{x})}^{5}$ by $1$ .

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

Step 4.52

Apply the product rule to $- \sqrt{x}$ .

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

Step 4.53

Raise $- 1$ to the power of $5$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + 5 x - {\sqrt{x}}^{5}$

Step 4.54

Rewrite ${\sqrt{x}}^{5}$ as $\sqrt{x^{5}}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + 5 x - \sqrt{x^{5}}$

Step 4.55

Rewrite $x^{5}$ as ${(x^{2})}^{2} x$ .

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Step 4.55.1

Factor out $x^{4}$ .

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + 5 x - \sqrt{x^{4} x}$

Step 4.55.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 4.56

Pull terms out from under the radical.

$\frac{1}{x^{5}} - \frac{5 \sqrt{x}}{x^{4}} + \frac{10}{x^{2}} - \frac{10 \sqrt{x}}{x} + 5 x - x^{2} \sqrt{x}$