Calculus Examples

$g (t) = \frac{t - \sqrt{t}}{t^{\frac{1}{3}}}$

Step 1

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

$\frac{d}{d t} [\frac{t - t^{\frac{1}{2}}}{t^{\frac{1}{3}}}]$

Step 2

Factor $t^{\frac{1}{2}}$ out of $t - t^{\frac{1}{2}}$ .

Tap for more steps...

Step 2.1

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

$\frac{d}{d t} [\frac{t^{1} - t^{\frac{1}{2}}}{t^{\frac{1}{3}}}]$

Step 2.2

Factor $t^{\frac{1}{2}}$ out of $t^{1}$ .

$\frac{d}{d t} [\frac{t^{\frac{1}{2}} t^{\frac{1}{2}} - t^{\frac{1}{2}}}{t^{\frac{1}{3}}}]$

Step 2.3

Factor $t^{\frac{1}{2}}$ out of $- t^{\frac{1}{2}}$ .

$\frac{d}{d t} [\frac{t^{\frac{1}{2}} t^{\frac{1}{2}} + t^{\frac{1}{2}} \cdot - 1}{t^{\frac{1}{3}}}]$

Step 2.4

Factor $t^{\frac{1}{2}}$ out of $t^{\frac{1}{2}} t^{\frac{1}{2}} + t^{\frac{1}{2}} \cdot - 1$ .

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

Step 3

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

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

Tap for more steps...

Step 4.5.1

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

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

Step 4.5.2

Multiply $3$ by $2$ .

$\frac{d}{d t} [t^{- \frac{2}{6} + \frac{1}{2} \cdot \frac{3}{3}} (t^{\frac{1}{2}} - 1)]$

Step 4.5.3

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

$\frac{d}{d t} [t^{- \frac{2}{6} + \frac{3}{2 \cdot 3}} (t^{\frac{1}{2}} - 1)]$

Step 4.5.4

Multiply $2$ by $3$ .

$\frac{d}{d t} [t^{- \frac{2}{6} + \frac{3}{6}} (t^{\frac{1}{2}} - 1)]$

Step 4.6

Combine the numerators over the common denominator.

$\frac{d}{d t} [t^{\frac{- 2 + 3}{6}} (t^{\frac{1}{2}} - 1)]$

Step 4.7

Add $- 2$ and $3$ .

$\frac{d}{d t} [t^{\frac{1}{6}} (t^{\frac{1}{2}} - 1)]$

Step 5

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = t^{\frac{1}{6}}$ and $g (t) = t^{\frac{1}{2}} - 1$ .

$t^{\frac{1}{6}} \frac{d}{d t} [t^{\frac{1}{2}} - 1] + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 6

Differentiate.

Tap for more steps...

Step 6.1

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

$t^{\frac{1}{6}} (\frac{d}{d t} [t^{\frac{1}{2}}] + \frac{d}{d t} [- 1]) + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 6.2

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

$t^{\frac{1}{6}} (\frac{1}{2} t^{\frac{1}{2} - 1} + \frac{d}{d t} [- 1]) + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 7

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

$t^{\frac{1}{6}} (\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d t} [- 1]) + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 8

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

$t^{\frac{1}{6}} (\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d t} [- 1]) + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 9

Combine the numerators over the common denominator.

$t^{\frac{1}{6}} (\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d t} [- 1]) + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 10

Simplify the numerator.

Tap for more steps...

Step 10.1

Multiply $- 1$ by $2$ .

$t^{\frac{1}{6}} (\frac{1}{2} t^{\frac{1 - 2}{2}} + \frac{d}{d t} [- 1]) + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 10.2

Subtract $2$ from $1$ .

$t^{\frac{1}{6}} (\frac{1}{2} t^{\frac{- 1}{2}} + \frac{d}{d t} [- 1]) + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 11

Combine fractions.

Tap for more steps...

Step 11.1

Move the negative in front of the fraction.

$t^{\frac{1}{6}} (\frac{1}{2} t^{- \frac{1}{2}} + \frac{d}{d t} [- 1]) + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 11.2

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

$t^{\frac{1}{6}} (\frac{t^{- \frac{1}{2}}}{2} + \frac{d}{d t} [- 1]) + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 11.3

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

$t^{\frac{1}{6}} (\frac{1}{2 t^{\frac{1}{2}}} + \frac{d}{d t} [- 1]) + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 12

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

$t^{\frac{1}{6}} (\frac{1}{2 t^{\frac{1}{2}}} + 0) + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 13

Combine fractions.

Tap for more steps...

Step 13.1

Add $\frac{1}{2 t^{\frac{1}{2}}}$ and $0$ .

$t^{\frac{1}{6}} \frac{1}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 13.2

Combine $t^{\frac{1}{6}}$ and $\frac{1}{2 t^{\frac{1}{2}}}$ .

$\frac{t^{\frac{1}{6}}}{2 t^{\frac{1}{2}}} + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 13.3

Move $t^{\frac{1}{6}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{1}{2 t^{\frac{1}{2}} t^{- \frac{1}{6}}} + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 14

Multiply $t^{\frac{1}{2}}$ by $t^{- \frac{1}{6}}$ by adding the exponents.

Tap for more steps...

Step 14.1

Move $t^{- \frac{1}{6}}$ .

$\frac{1}{2 (t^{- \frac{1}{6}} t^{\frac{1}{2}})} + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 14.2

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

$\frac{1}{2 t^{- \frac{1}{6} + \frac{1}{2}}} + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 14.3

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

$\frac{1}{2 t^{- \frac{1}{6} + \frac{1}{2} \cdot \frac{3}{3}}} + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 14.4

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

Tap for more steps...

Step 14.4.1

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

$\frac{1}{2 t^{- \frac{1}{6} + \frac{3}{2 \cdot 3}}} + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 14.4.2

Multiply $2$ by $3$ .

$\frac{1}{2 t^{- \frac{1}{6} + \frac{3}{6}}} + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 14.5

Combine the numerators over the common denominator.

$\frac{1}{2 t^{\frac{- 1 + 3}{6}}} + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 14.6

Add $- 1$ and $3$ .

$\frac{1}{2 t^{\frac{2}{6}}} + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 14.7

Cancel the common factor of $2$ and $6$ .

Tap for more steps...

Step 14.7.1

Factor $2$ out of $2$ .

$\frac{1}{2 t^{\frac{2 (1)}{6}}} + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 14.7.2

Cancel the common factors.

Tap for more steps...

Step 14.7.2.1

Factor $2$ out of $6$ .

$\frac{1}{2 t^{\frac{2 \cdot 1}{2 \cdot 3}}} + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 14.7.2.2

Cancel the common factor.

$\frac{1}{2 t^{\frac{2 \cdot 1}{2 \cdot 3}}} + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 14.7.2.3

Rewrite the expression.

$\frac{1}{2 t^{\frac{1}{3}}} + (t^{\frac{1}{2}} - 1) \frac{d}{d t} [t^{\frac{1}{6}}]$

Step 15

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{1}{6}$ .

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

Step 16

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

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

Step 17

Combine $- 1$ and $\frac{6}{6}$ .

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

Step 18

Combine the numerators over the common denominator.

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

Step 19

Simplify the numerator.

Tap for more steps...

Step 19.1

Multiply $- 1$ by $6$ .

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

Step 19.2

Subtract $6$ from $1$ .

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

Step 20

Move the negative in front of the fraction.

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

Step 21

Combine $\frac{1}{6}$ and $t^{- \frac{5}{6}}$ .

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

Step 22

Move $t^{- \frac{5}{6}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 23

To write $(t^{\frac{1}{2}} - 1) \frac{1}{6 t^{\frac{5}{6}}}$ as a fraction with a common denominator, multiply by $\frac{2 t^{\frac{1}{3}}}{2 t^{\frac{1}{3}}}$ .

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

Step 24

Combine $(t^{\frac{1}{2}} - 1) \frac{1}{6 t^{\frac{5}{6}}}$ and $\frac{2 t^{\frac{1}{3}}}{2 t^{\frac{1}{3}}}$ .

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

Step 25

Combine the numerators over the common denominator.

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

Step 26

Combine $2$ and $\frac{1}{6 t^{\frac{5}{6}}}$ .

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

Step 27

Combine $t^{\frac{1}{3}}$ and $\frac{2}{6 t^{\frac{5}{6}}}$ .

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

Step 28

Simplify the expression.

Tap for more steps...

Step 28.1

Move $2$ to the left of $t^{\frac{1}{3}}$ .

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

Step 28.2

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

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

Step 29

Multiply $t^{\frac{5}{6}}$ by $t^{- \frac{1}{3}}$ by adding the exponents.

Tap for more steps...

Step 29.1

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

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

Step 29.2

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

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

Step 29.3

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

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

Step 29.4

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

Tap for more steps...

Step 29.4.1

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

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

Step 29.4.2

Multiply $3$ by $2$ .

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

Step 29.5

Combine the numerators over the common denominator.

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

Step 29.6

Add $- 2$ and $5$ .

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

Step 29.7

Cancel the common factor of $3$ and $6$ .

Tap for more steps...

Step 29.7.1

Factor $3$ out of $3$ .

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

Step 29.7.2

Cancel the common factors.

Tap for more steps...

Step 29.7.2.1

Factor $3$ out of $6$ .

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

Step 29.7.2.2

Cancel the common factor.

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

Step 29.7.2.3

Rewrite the expression.

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

Step 30

Factor $2$ out of $2$ .

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

Step 31

Cancel the common factors.

Tap for more steps...

Step 31.1

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

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

Step 31.2

Cancel the common factor.

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

Step 31.3

Rewrite the expression.

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

Step 32

Simplify.

Tap for more steps...

Step 32.1

Apply the distributive property.

$\frac{1 + t^{\frac{1}{2}} \frac{1}{3 t^{\frac{1}{2}}} - 1 \frac{1}{3 t^{\frac{1}{2}}}}{2 t^{\frac{1}{3}}}$

Step 32.2

Simplify the numerator.

Tap for more steps...

Step 32.2.1

Simplify each term.

Tap for more steps...

Step 32.2.1.1

Cancel the common factor of $t^{\frac{1}{2}}$ .

Tap for more steps...

Step 32.2.1.1.1

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

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

Step 32.2.1.1.2

Cancel the common factor.

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

Step 32.2.1.1.3

Rewrite the expression.

$\frac{1 + \frac{1}{3} - 1 \frac{1}{3 t^{\frac{1}{2}}}}{2 t^{\frac{1}{3}}}$

Step 32.2.1.2

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

$\frac{1 + \frac{1}{3} - \frac{1}{3 t^{\frac{1}{2}}}}{2 t^{\frac{1}{3}}}$

Step 32.2.2

Write $1$ as a fraction with a common denominator.

$\frac{\frac{3}{3} + \frac{1}{3} - \frac{1}{3 t^{\frac{1}{2}}}}{2 t^{\frac{1}{3}}}$

Step 32.2.3

Combine the numerators over the common denominator.

$\frac{\frac{3 + 1}{3} - \frac{1}{3 t^{\frac{1}{2}}}}{2 t^{\frac{1}{3}}}$

Step 32.2.4

Add $3$ and $1$ .

$\frac{\frac{4}{3} - \frac{1}{3 t^{\frac{1}{2}}}}{2 t^{\frac{1}{3}}}$

Step 32.3

Combine terms.

Tap for more steps...

Step 32.3.1

Multiply $\frac{\frac{4}{3} - \frac{1}{3 t^{\frac{1}{2}}}}{2 t^{\frac{1}{3}}}$ by $\frac{3}{3}$ .

$\frac{3}{3} \cdot \frac{\frac{4}{3} - \frac{1}{3 t^{\frac{1}{2}}}}{2 t^{\frac{1}{3}}}$

Step 32.3.2

Combine.

$\frac{3 (\frac{4}{3} - \frac{1}{3 t^{\frac{1}{2}}})}{3 (2 t^{\frac{1}{3}})}$

Step 32.3.3

Apply the distributive property.

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

Step 32.3.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 32.3.4.1

Cancel the common factor.

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

Step 32.3.4.2

Rewrite the expression.

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

Step 32.3.5

Multiply $- 1$ by $3$ .

$\frac{4 - 3 \frac{1}{3 t^{\frac{1}{2}}}}{3 (2 t^{\frac{1}{3}})}$

Step 32.3.6

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

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

Step 32.3.7

Factor $3$ out of $- 3$ .

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

Step 32.3.8

Cancel the common factors.

Tap for more steps...

Step 32.3.8.1

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

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

Step 32.3.8.2

Cancel the common factor.

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

Step 32.3.8.3

Rewrite the expression.

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

Step 32.3.9

Move the negative in front of the fraction.

$\frac{4 - \frac{1}{t^{\frac{1}{2}}}}{3 (2 t^{\frac{1}{3}})}$

Step 32.3.10

Multiply $2$ by $3$ .

$\frac{4 - \frac{1}{t^{\frac{1}{2}}}}{6 t^{\frac{1}{3}}}$

Step 32.4

Simplify the numerator.

Tap for more steps...

Step 32.4.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{t^{\frac{1}{2}}}{t^{\frac{1}{2}}}$ .

$\frac{\frac{4 t^{\frac{1}{2}}}{t^{\frac{1}{2}}} - \frac{1}{t^{\frac{1}{2}}}}{6 t^{\frac{1}{3}}}$

Step 32.4.2

Combine the numerators over the common denominator.

$\frac{\frac{4 t^{\frac{1}{2}} - 1}{t^{\frac{1}{2}}}}{6 t^{\frac{1}{3}}}$

Step 32.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 t^{\frac{1}{2}} - 1}{t^{\frac{1}{2}}} \cdot \frac{1}{6 t^{\frac{1}{3}}}$

Step 32.6

Multiply $\frac{4 t^{\frac{1}{2}} - 1}{t^{\frac{1}{2}}} \cdot \frac{1}{6 t^{\frac{1}{3}}}$ .

Tap for more steps...

Step 32.6.1

Multiply $\frac{4 t^{\frac{1}{2}} - 1}{t^{\frac{1}{2}}}$ by $\frac{1}{6 t^{\frac{1}{3}}}$ .

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

Step 32.6.2

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

Tap for more steps...

Step 32.6.2.1

Move $t^{\frac{1}{3}}$ .

$\frac{4 t^{\frac{1}{2}} - 1}{t^{\frac{1}{3}} t^{\frac{1}{2}} \cdot 6}$

Step 32.6.2.2

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

$\frac{4 t^{\frac{1}{2}} - 1}{t^{\frac{1}{3} + \frac{1}{2}} \cdot 6}$

Step 32.6.2.3

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

$\frac{4 t^{\frac{1}{2}} - 1}{t^{\frac{1}{3} \cdot \frac{2}{2} + \frac{1}{2}} \cdot 6}$

Step 32.6.2.4

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

$\frac{4 t^{\frac{1}{2}} - 1}{t^{\frac{1}{3} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{3}{3}} \cdot 6}$

Step 32.6.2.5

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

Tap for more steps...

Step 32.6.2.5.1

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

$\frac{4 t^{\frac{1}{2}} - 1}{t^{\frac{2}{3 \cdot 2} + \frac{1}{2} \cdot \frac{3}{3}} \cdot 6}$

Step 32.6.2.5.2

Multiply $3$ by $2$ .

$\frac{4 t^{\frac{1}{2}} - 1}{t^{\frac{2}{6} + \frac{1}{2} \cdot \frac{3}{3}} \cdot 6}$

Step 32.6.2.5.3

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

$\frac{4 t^{\frac{1}{2}} - 1}{t^{\frac{2}{6} + \frac{3}{2 \cdot 3}} \cdot 6}$

Step 32.6.2.5.4

Multiply $2$ by $3$ .

$\frac{4 t^{\frac{1}{2}} - 1}{t^{\frac{2}{6} + \frac{3}{6}} \cdot 6}$

Step 32.6.2.6

Combine the numerators over the common denominator.

$\frac{4 t^{\frac{1}{2}} - 1}{t^{\frac{2 + 3}{6}} \cdot 6}$

Step 32.6.2.7

Add $2$ and $3$ .

$\frac{4 t^{\frac{1}{2}} - 1}{t^{\frac{5}{6}} \cdot 6}$

Step 32.7

Move $6$ to the left of $t^{\frac{5}{6}}$ .

$\frac{4 t^{\frac{1}{2}} - 1}{6 t^{\frac{5}{6}}}$