Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Add $- 1$ and $3$ .

$g t^{\frac{2}{3}} (t - \sqrt{t})$

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

Move $t$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

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

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 4.5

Add $3$ and $2$ .

$g t^{\frac{5}{3}} + g t^{\frac{2}{3}} (- \sqrt{t})$

Step 5

Rewrite using the commutative property of multiplication.

$g t^{\frac{5}{3}} - g t^{\frac{2}{3}} \sqrt{t}$

Step 6

Multiply $- g t^{\frac{2}{3}} \sqrt{t}$ .

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Step 6.1

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

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

Step 6.2

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

$g t^{\frac{5}{3}} - g t^{\frac{1}{2} + \frac{2}{3}}$

Step 6.3

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

$g t^{\frac{5}{3}} - g t^{\frac{1}{2} \cdot \frac{3}{3} + \frac{2}{3}}$

Step 6.4

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

$g t^{\frac{5}{3}} - g t^{\frac{1}{2} \cdot \frac{3}{3} + \frac{2}{3} \cdot \frac{2}{2}}$

Step 6.5

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

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Step 6.5.1

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

$g t^{\frac{5}{3}} - g t^{\frac{3}{2 \cdot 3} + \frac{2}{3} \cdot \frac{2}{2}}$

Step 6.5.2

Multiply $2$ by $3$ .

$g t^{\frac{5}{3}} - g t^{\frac{3}{6} + \frac{2}{3} \cdot \frac{2}{2}}$

Step 6.5.3

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

$g t^{\frac{5}{3}} - g t^{\frac{3}{6} + \frac{2 \cdot 2}{3 \cdot 2}}$

Step 6.5.4

Multiply $3$ by $2$ .

$g t^{\frac{5}{3}} - g t^{\frac{3}{6} + \frac{2 \cdot 2}{6}}$

Step 6.6

Combine the numerators over the common denominator.

$g t^{\frac{5}{3}} - g t^{\frac{3 + 2 \cdot 2}{6}}$

Step 6.7

Simplify the numerator.

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Step 6.7.1

Multiply $2$ by $2$ .

$g t^{\frac{5}{3}} - g t^{\frac{3 + 4}{6}}$

Step 6.7.2

Add $3$ and $4$ .

$g t^{\frac{5}{3}} - g t^{\frac{7}{6}}$