Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Move the negative in front of the fraction.

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

Combine the numerators over the common denominator.

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

Step 4.1.4

Add $1$ and $1$ .

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

Step 4.1.5

Divide $2$ by $2$ .

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

Step 4.2

Simplify $2 {(x - 1)}^{1}$ .

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Multiply $2$ by $- 1$ .

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

Step 4.5

Add $x$ and $2 x$ .

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