Question

1. rationalize the numerator of each of the following expressions:

a. \\(\frac{\sqrt{a} - 2}{a - 4}\\)
b. \\(\frac{\sqrt{x + 4} - 2}{x}\\)
c. \\(\frac{\sqrt{x + h} - x}{x}\\)

Explanation:

Part (a)

Step1: Identify conjugate of numerator

The numerator is $\sqrt{a} - 2$, so its conjugate is $\sqrt{a} + 2$. Multiply numerator and denominator by $\sqrt{a} + 2$.
$$\frac{\sqrt{a} - 2}{a - 4} \times \frac{\sqrt{a} + 2}{\sqrt{a} + 2}$$

Step2: Expand numerator and denominator

Numerator: $(\sqrt{a} - 2)(\sqrt{a} + 2) = (\sqrt{a})^2 - 2^2 = a - 4$ (using difference of squares: $(x - y)(x + y) = x^2 - y^2$)
Denominator: $(a - 4)(\sqrt{a} + 2)$

Step3: Simplify the fraction

Now we have $\frac{a - 4}{(a - 4)(\sqrt{a} + 2)}$. Assuming $a
eq 4$ (to avoid division by zero), we can cancel out $(a - 4)$ from numerator and denominator.
$$\frac{1}{\sqrt{a} + 2}$$

Step1: Identify conjugate of numerator

The numerator is $\sqrt{x + 4} - 2$, so its conjugate is $\sqrt{x + 4} + 2$. Multiply numerator and denominator by $\sqrt{x + 4} + 2$.
$$\frac{\sqrt{x + 4} - 2}{x} \times \frac{\sqrt{x + 4} + 2}{\sqrt{x + 4} + 2}$$

Step2: Expand numerator and denominator

Numerator: $(\sqrt{x + 4} - 2)(\sqrt{x + 4} + 2) = (\sqrt{x + 4})^2 - 2^2 = (x + 4) - 4 = x$ (using difference of squares)
Denominator: $x(\sqrt{x + 4} + 2)$

Step3: Simplify the fraction

Now we have $\frac{x}{x(\sqrt{x + 4} + 2)}$. Assuming $x
eq 0$ (to avoid division by zero), we can cancel out $x$ from numerator and denominator.
$$\frac{1}{\sqrt{x + 4} + 2}$$

Step1: Identify conjugate of numerator

The numerator is $\sqrt{x + h} - \sqrt{x}$ (I assume there was a typo and it should be $\sqrt{x + h} - \sqrt{x}$ instead of $\sqrt{x + h} - x$; if it's $\sqrt{x + h} - x$, the process is similar but let's proceed with the likely intended expression $\sqrt{x + h} - \sqrt{x}$). The conjugate of $\sqrt{x + h} - \sqrt{x}$ is $\sqrt{x + h} + \sqrt{x}$. Multiply numerator and denominator by $\sqrt{x + h} + \sqrt{x}$.
$$\frac{\sqrt{x + h} - \sqrt{x}}{x} \times \frac{\sqrt{x + h} + \sqrt{x}}{\sqrt{x + h} + \sqrt{x}}$$

Step2: Expand numerator and denominator

Numerator: $(\sqrt{x + h} - \sqrt{x})(\sqrt{x + h} + \sqrt{x}) = (x + h) - x = h$ (using difference of squares)
Denominator: $x(\sqrt{x + h} + \sqrt{x})$

Step3: Simplify the fraction

The simplified form is $\frac{h}{x(\sqrt{x + h} + \sqrt{x})}$

If we take the original numerator as $\sqrt{x + h} - x$ (as given), the conjugate of $\sqrt{x + h} - x$ is $\sqrt{x + h} + x$. Multiply numerator and denominator by $\sqrt{x + h} + x$:

Step1 (corrected for original numerator):

$$\frac{\sqrt{x + h} - x}{x} \times \frac{\sqrt{x + h} + x}{\sqrt{x + h} + x}$$

Step2 (corrected):

Numerator: $(\sqrt{x + h})^2 - x^2 = (x + h) - x^2 = -x^2 + x + h$
Denominator: $x(\sqrt{x + h} + x)$
This seems more complicated, so likely a typo. Assuming the intended numerator was $\sqrt{x + h} - \sqrt{x}$, the simplified form is $\frac{h}{x(\sqrt{x + h} + \sqrt{x})}$.

Answer:

$\frac{1}{\sqrt{a} + 2}$ (for $a
eq 4$)

Part (b)