Precalculus Examples

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

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Combine the numerators over the common denominator.

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

Step 5.1.3

Add $1$ and $1$ .

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

Step 5.1.4

Divide $2$ by $2$ .

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

Step 5.2

Simplify $x^{1}$ .

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$x - 2$