Calculus Examples

2 \sqrt{x} + 6 x^{\frac{3}{2}}

$2 \sqrt{x} + 6 x^{\frac{3}{2}}$

Step 1

Factor out the GCF of

2 x

$2 x$ from

2 \sqrt{x} + 6 x^{\frac{3}{2}}

$2 \sqrt{x} + 6 x^{\frac{3}{2}}$ .

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Step 1.1

Factor out the GCF of

2 x

$2 x$ from each term in the polynomial.

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Step 1.1.1

Factor out the GCF of

2 x

$2 x$ from the expression

2 \sqrt{x}

$2 \sqrt{x}$ .

2 x (\frac{\sqrt{x}}{x}) + 6 x^{\frac{3}{2}}

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

Step 1.1.2

Factor out the GCF of

2 x

$2 x$ from the expression

6 x^{\frac{3}{2}}

$6 x^{\frac{3}{2}}$ .

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

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

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

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

Step 1.2

Since all the terms share a common factor of

2 x

$2 x$ , it can be factored out of each term.

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

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

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

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

Step 2

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

x^{\frac{1}{2}}

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

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

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

Step 3

Factor

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ out of

\frac{x^{\frac{1}{2}}}{x} + 3 x^{\frac{1}{2}}

$\frac{x^{\frac{1}{2}}}{x} + 3 x^{\frac{1}{2}}$ .

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Step 3.1

Factor

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ out of

\frac{x^{\frac{1}{2}}}{x}

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

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

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

Step 3.2

Factor

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ out of

3 x^{\frac{1}{2}}

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

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

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

Step 3.3

Factor

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ out of

x^{\frac{1}{2}} \frac{1}{x} + x^{\frac{1}{2}} \cdot 3

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

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

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

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

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

Step 4

To write

3

$3$ as a fraction with a common denominator, multiply by

\frac{x}{x}

$\frac{x}{x}$ .

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

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

Step 5

Combine the numerators over the common denominator.

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

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