Precalculus Examples

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

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Rewrite $\frac{1}{x^{\frac{1}{2}}} - 2 x^{\frac{1}{2}} + x^{\frac{3}{2}}$ in a factored form.

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Step 2.1

Let $u = x^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{2}}$ .

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

Step 2.2

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

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Step 2.2.1

Reorder $\frac{1}{u}$ and $- 2 u$ .

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

Step 2.2.2

Factor $- 1$ out of $- 2 u$ .

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

Step 2.2.3

Factor $- 1$ out of $\frac{1}{u}$ .

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

Step 2.2.4

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

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

Step 2.2.5

Factor $- 1$ out of $- (2 u) - \frac{- 1}{u}$ .

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

Step 2.2.6

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

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

Step 2.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

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

Step 2.4

Move the negative in front of the fraction.

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

Step 2.5

Reorder terms.

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