Questions: Rationalize the denominator of (sqrt(x))/(sqrt(x)+sqrt(a)). Assume that all variables represent positive real numbers.

Solution Steps

To rationalize the denominator of the expression

\[ \frac{\sqrt{x}}{\sqrt{x}+\sqrt{a}}, \]

we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{x} - \sqrt{a}\). This gives us:

\[ \frac{\sqrt{x}(\sqrt{x} - \sqrt{a})}{(\sqrt{x}+\sqrt{a})(\sqrt{x}-\sqrt{a})}. \]

Next, we simplify the expression. The denominator simplifies using the difference of squares:

\[ (\sqrt{x}+\sqrt{a})(\sqrt{x}-\sqrt{a}) = x - a. \]

Thus, the expression becomes:

\[ \frac{\sqrt{x}(\sqrt{x} - \sqrt{a})}{x - a}. \]

The numerator can be expressed as:

\[ \sqrt{x}(\sqrt{x} - \sqrt{a}) = x - \sqrt{a}\sqrt{x}. \]

Therefore, the fully simplified expression is:

\[ \frac{x - \sqrt{a}\sqrt{x}}{x - a}. \]

Final Answer