Question

rationalize the denominator of \\(\frac{\sqrt{a} - 1}{4\sqrt{a} - \sqrt{b}}\\). assume that all variables represent positive real numbers.

Explanation:

Step1: Identify the conjugate of the denominator

The denominator is \(4\sqrt{a}-\sqrt{b}\), so its conjugate is \(4\sqrt{a}+\sqrt{b}\). We multiply the numerator and the denominator by this conjugate.
\[
\frac{\sqrt{a} - 1}{4\sqrt{a}-\sqrt{b}}\times\frac{4\sqrt{a}+\sqrt{b}}{4\sqrt{a}+\sqrt{b}}
\]

Step2: Multiply the numerators

Using the distributive property (FOIL method) for the numerators: \((\sqrt{a}-1)(4\sqrt{a}+\sqrt{b}) = \sqrt{a}\times4\sqrt{a}+\sqrt{a}\times\sqrt{b}-1\times4\sqrt{a}-1\times\sqrt{b}\)
Simplify each term:
\(\sqrt{a}\times4\sqrt{a}=4a\), \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\), \(-1\times4\sqrt{a}=-4\sqrt{a}\), \(-1\times\sqrt{b}=-\sqrt{b}\)
So the numerator becomes \(4a+\sqrt{ab}-4\sqrt{a}-\sqrt{b}\)

Step3: Multiply the denominators

Using the difference of squares formula \((x - y)(x + y)=x^{2}-y^{2}\), where \(x = 4\sqrt{a}\) and \(y=\sqrt{b}\)
\[
(4\sqrt{a})^{2}-(\sqrt{b})^{2}=16a - b
\]

Step4: Combine the results

Put the simplified numerator and denominator together:
\[
\frac{4a+\sqrt{ab}-4\sqrt{a}-\sqrt{b}}{16a - b}
\]

Answer:

\(\frac{4a+\sqrt{ab}-4\sqrt{a}-\sqrt{b}}{16a - b}\)