Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Write each expression with a common denominator of $x^{\frac{1}{2}} {(x + 2)}^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{{(x + 2)}^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ by $\frac{{(x + 2)}^{\frac{1}{2}}}{{(x + 2)}^{\frac{1}{2}}}$ .

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

Step 7.2

Multiply $\frac{x^{\frac{1}{2}}}{{(x + 2)}^{\frac{1}{2}}}$ by $\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ .

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

Step 7.3

Reorder the factors of ${(x + 2)}^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

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

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Step 9.1

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

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Step 9.1.1

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

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

Step 9.1.2

Combine the numerators over the common denominator.

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

Step 9.1.3

Add $1$ and $1$ .

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

Step 9.1.4

Divide $2$ by $2$ .

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

Step 9.2

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

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

Step 9.3

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

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Step 9.3.1

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

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

Step 9.3.2

Combine the numerators over the common denominator.

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

Step 9.3.3

Add $1$ and $1$ .

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

Step 9.3.4

Divide $2$ by $2$ .

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

Step 9.4

Simplify $x^{1}$ .

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

Step 9.5

Add $x$ and $x$ .

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

Step 9.6

Factor $2$ out of $2 x + 2$ .

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Step 9.6.1

Factor $2$ out of $2 x$ .

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

Step 9.6.2

Factor $2$ out of $2$ .

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

Step 9.6.3

Factor $2$ out of $2 (x) + 2 (1)$ .

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