Calculus Examples

$\frac{5 \sqrt{x} - 2 \sqrt[3]{x}}{4 x}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

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

$\frac{5 x^{\frac{1}{2}} - 2 \sqrt[3]{x}}{4 x}$

Step 1.2

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

$\frac{5 x^{\frac{1}{2}} - 2 x^{\frac{1}{3}}}{4 x}$

Step 1.3

Factor $x^{\frac{1}{3}}$ out of $5 x^{\frac{1}{2}} - 2 x^{\frac{1}{3}}$ .

Tap for more steps...

Step 1.3.1

Factor $x^{\frac{1}{3}}$ out of $5 x^{\frac{1}{2}}$ .

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

Step 1.3.2

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

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

Step 1.3.3

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{1}{3}} (5 x^{\frac{1}{6}}) + x^{\frac{1}{3}} \cdot - 2$ .

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

Step 2

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

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Add $- 1$ and $3$ .

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