Calculus Examples

$(1 - \sqrt{x}) (1 - \sqrt[3]{x})$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

$\frac{d}{d x} [(1 - x^{\frac{1}{2}}) (1 - \sqrt[3]{x})]$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\frac{d}{d x} [(1 - x^{\frac{1}{2}}) (1 - x^{\frac{1}{3}})]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 1 - x^{\frac{1}{2}}$ and $g (x) = 1 - x^{\frac{1}{3}}$ .

$(1 - x^{\frac{1}{2}}) \frac{d}{d x} [1 - x^{\frac{1}{3}}] + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [1 - x^{\frac{1}{2}}]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $1 - x^{\frac{1}{3}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x^{\frac{1}{3}}]$ .

$(1 - x^{\frac{1}{2}}) (\frac{d}{d x} [1] + \frac{d}{d x} [- x^{\frac{1}{3}}]) + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [1 - x^{\frac{1}{2}}]$

Step 3.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(1 - x^{\frac{1}{2}}) (0 + \frac{d}{d x} [- x^{\frac{1}{3}}]) + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [1 - x^{\frac{1}{2}}]$

Step 3.3

Add $0$ and $\frac{d}{d x} [- x^{\frac{1}{3}}]$ .

$(1 - x^{\frac{1}{2}}) \frac{d}{d x} [- x^{\frac{1}{3}}] + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [1 - x^{\frac{1}{2}}]$

Step 3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{1}{3}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{1}{3}}]$ .

$(1 - x^{\frac{1}{2}}) (- \frac{d}{d x} [x^{\frac{1}{3}}]) + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [1 - x^{\frac{1}{2}}]$

Step 3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{3}$ .

$(1 - x^{\frac{1}{2}}) (- (\frac{1}{3} x^{\frac{1}{3} - 1})) + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [1 - x^{\frac{1}{2}}]$

Step 4

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

$(1 - x^{\frac{1}{2}}) (- (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}})) + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [1 - x^{\frac{1}{2}}]$

Step 5

Combine $- 1$ and $\frac{3}{3}$ .

$(1 - x^{\frac{1}{2}}) (- (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}})) + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [1 - x^{\frac{1}{2}}]$

Step 6

Combine the numerators over the common denominator.

$(1 - x^{\frac{1}{2}}) (- (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}})) + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [1 - x^{\frac{1}{2}}]$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $3$ .

$(1 - x^{\frac{1}{2}}) (- (\frac{1}{3} x^{\frac{1 - 3}{3}})) + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [1 - x^{\frac{1}{2}}]$

Step 7.2

Subtract $3$ from $1$ .

$(1 - x^{\frac{1}{2}}) (- (\frac{1}{3} x^{\frac{- 2}{3}})) + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [1 - x^{\frac{1}{2}}]$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$(1 - x^{\frac{1}{2}}) (- (\frac{1}{3} x^{- \frac{2}{3}})) + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [1 - x^{\frac{1}{2}}]$

Step 8.2

Combine $\frac{1}{3}$ and $x^{- \frac{2}{3}}$ .

$(1 - x^{\frac{1}{2}}) (- \frac{x^{- \frac{2}{3}}}{3}) + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [1 - x^{\frac{1}{2}}]$

Step 8.3

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

$(1 - x^{\frac{1}{2}}) (- \frac{1}{3 x^{\frac{2}{3}}}) + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [1 - x^{\frac{1}{2}}]$

Step 9

By the Sum Rule, the derivative of $1 - x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x^{\frac{1}{2}}]$ .

$(1 - x^{\frac{1}{2}}) (- \frac{1}{3 x^{\frac{2}{3}}}) + (1 - x^{\frac{1}{3}}) (\frac{d}{d x} [1] + \frac{d}{d x} [- x^{\frac{1}{2}}])$

Step 10

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(1 - x^{\frac{1}{2}}) (- \frac{1}{3 x^{\frac{2}{3}}}) + (1 - x^{\frac{1}{3}}) (0 + \frac{d}{d x} [- x^{\frac{1}{2}}])$

Step 11

Add $0$ and $\frac{d}{d x} [- x^{\frac{1}{2}}]$ .

$(1 - x^{\frac{1}{2}}) (- \frac{1}{3 x^{\frac{2}{3}}}) + (1 - x^{\frac{1}{3}}) \frac{d}{d x} [- x^{\frac{1}{2}}]$

Step 12

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$(1 - x^{\frac{1}{2}}) (- \frac{1}{3 x^{\frac{2}{3}}}) + (1 - x^{\frac{1}{3}}) (- \frac{d}{d x} [x^{\frac{1}{2}}])$

Step 13

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 14

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

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

Step 15

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

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

Step 16

Combine the numerators over the common denominator.

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

Step 17

Simplify the numerator.

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Step 17.1

Multiply $- 1$ by $2$ .

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

Step 17.2

Subtract $2$ from $1$ .

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

Step 18

Move the negative in front of the fraction.

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

Step 19

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

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

Step 20

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

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

Step 21

Simplify.

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Step 21.1

Apply the distributive property.

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

Step 21.2

Apply the distributive property.

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

Step 21.3

Combine terms.

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Step 21.3.1

Multiply $- 1$ by $1$ .

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

Step 21.3.2

Rewrite $- 1 \frac{1}{3 x^{\frac{2}{3}}}$ as $- \frac{1}{3 x^{\frac{2}{3}}}$ .

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

Step 21.3.3

Multiply $- 1$ by $- 1$ .

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

Step 21.3.4

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

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

Step 21.3.5

Combine $x^{\frac{1}{2}}$ and $\frac{1}{3 x^{\frac{2}{3}}}$ .

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

Step 21.3.6

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

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

Step 21.3.7

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

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Step 21.3.7.1

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

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

Step 21.3.7.2

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

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

Step 21.3.7.3

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 21.3.7.4

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

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

Step 21.3.7.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 21.3.7.5.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

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

Step 21.3.7.5.2

Multiply $2$ by $3$ .

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

Step 21.3.7.5.3

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

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

Step 21.3.7.5.4

Multiply $3$ by $2$ .

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

Step 21.3.7.6

Combine the numerators over the common denominator.

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

Step 21.3.7.7

Simplify the numerator.

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Step 21.3.7.7.1

Multiply $2$ by $2$ .

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

Step 21.3.7.7.2

Add $- 3$ and $4$ .

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

Step 21.3.8

Multiply $- 1$ by $1$ .

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

Step 21.3.9

Rewrite $- 1 \frac{1}{2 x^{\frac{1}{2}}}$ as $- \frac{1}{2 x^{\frac{1}{2}}}$ .

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

Step 21.3.10

Multiply $- 1$ by $- 1$ .

$- \frac{1}{3 x^{\frac{2}{3}}} + \frac{1}{3 x^{\frac{1}{6}}} - \frac{1}{2 x^{\frac{1}{2}}} + 1 x^{\frac{1}{3}} \frac{1}{2 x^{\frac{1}{2}}}$

Step 21.3.11

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

$- \frac{1}{3 x^{\frac{2}{3}}} + \frac{1}{3 x^{\frac{1}{6}}} - \frac{1}{2 x^{\frac{1}{2}}} + x^{\frac{1}{3}} \frac{1}{2 x^{\frac{1}{2}}}$

Step 21.3.12

Combine $x^{\frac{1}{3}}$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$- \frac{1}{3 x^{\frac{2}{3}}} + \frac{1}{3 x^{\frac{1}{6}}} - \frac{1}{2 x^{\frac{1}{2}}} + \frac{x^{\frac{1}{3}}}{2 x^{\frac{1}{2}}}$

Step 21.3.13

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

$- \frac{1}{3 x^{\frac{2}{3}}} + \frac{1}{3 x^{\frac{1}{6}}} - \frac{1}{2 x^{\frac{1}{2}}} + \frac{1}{2 x^{\frac{1}{2}} x^{- \frac{1}{3}}}$

Step 21.3.14

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

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Step 21.3.14.1

Move $x^{- \frac{1}{3}}$ .

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

Step 21.3.14.2

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

$- \frac{1}{3 x^{\frac{2}{3}}} + \frac{1}{3 x^{\frac{1}{6}}} - \frac{1}{2 x^{\frac{1}{2}}} + \frac{1}{2 x^{- \frac{1}{3} + \frac{1}{2}}}$

Step 21.3.14.3

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 21.3.14.4

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 21.3.14.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 21.3.14.5.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

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

Step 21.3.14.5.2

Multiply $3$ by $2$ .

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

Step 21.3.14.5.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

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

Step 21.3.14.5.4

Multiply $2$ by $3$ .

$- \frac{1}{3 x^{\frac{2}{3}}} + \frac{1}{3 x^{\frac{1}{6}}} - \frac{1}{2 x^{\frac{1}{2}}} + \frac{1}{2 x^{- \frac{2}{6} + \frac{3}{6}}}$

Step 21.3.14.6

Combine the numerators over the common denominator.

$- \frac{1}{3 x^{\frac{2}{3}}} + \frac{1}{3 x^{\frac{1}{6}}} - \frac{1}{2 x^{\frac{1}{2}}} + \frac{1}{2 x^{\frac{- 2 + 3}{6}}}$

Step 21.3.14.7

Add $- 2$ and $3$ .

$- \frac{1}{3 x^{\frac{2}{3}}} + \frac{1}{3 x^{\frac{1}{6}}} - \frac{1}{2 x^{\frac{1}{2}}} + \frac{1}{2 x^{\frac{1}{6}}}$

Step 21.3.15

To write $\frac{1}{3 x^{\frac{1}{6}}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 21.3.16

To write $\frac{1}{2 x^{\frac{1}{6}}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 21.3.17

Write each expression with a common denominator of $6 x^{\frac{1}{6}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 21.3.17.1

Multiply $\frac{1}{3 x^{\frac{1}{6}}}$ by $\frac{2}{2}$ .

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

Step 21.3.17.2

Multiply $2$ by $3$ .

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

Step 21.3.17.3

Multiply $\frac{1}{2 x^{\frac{1}{6}}}$ by $\frac{3}{3}$ .

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

Step 21.3.17.4

Multiply $3$ by $2$ .

$- \frac{1}{3 x^{\frac{2}{3}}} + \frac{2}{6 x^{\frac{1}{6}}} + \frac{3}{6 x^{\frac{1}{6}}} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 21.3.18

Combine the numerators over the common denominator.

$- \frac{1}{3 x^{\frac{2}{3}}} + \frac{2 + 3}{6 x^{\frac{1}{6}}} - \frac{1}{2 x^{\frac{1}{2}}}$

Step 21.3.19

Add $2$ and $3$ .

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

Step 21.4

Reorder terms.

$\frac{5}{6 x^{\frac{1}{6}}} - \frac{1}{3 x^{\frac{2}{3}}} - \frac{1}{2 x^{\frac{1}{2}}}$