Chapter SH, Problem 6.20EP

To determine

Find the simplified form of the given expression.

Answer to Problem 6.20EP

The simplified form of the expression is $\frac{a-6a^{\frac{1}{2}}+9}{a-9}$ .

Explanation of Solution

Given:

The given expression is $\frac{a^{\frac{1}{2}}-3}{a^{\frac{1}{2}}+3}$ .

Calculation:

Multiply numerator and denominator by conjugate of $a^{\frac{1}{2}}+3$ .

  $\begin{array}{l}\frac{a^{\frac{1}{2}}-3}{a^{\frac{1}{2}}+3}\\ =\frac{a^{\frac{1}{2}}-3}{a^{\frac{1}{2}}+3}\cdot \frac{a^{\frac{1}{2}}-3}{a^{\frac{1}{2}}-3}\\ =\frac{a^{\frac{1}{2}}\cdot a^{\frac{1}{2}}-a^{\frac{1}{2}}\cdot 3-3\cdot a^{\frac{1}{2}}+3^2}{{\left(a^{\frac{1}{2}}\right)}^2-3^2}\\ =\frac{a^{\frac{1}{2}+\frac{1}{2}}-6a^{\frac{1}{2}}+9}{a^{\frac{1}{2}\cdot 2}-9}\\ =\frac{a-6a^{\frac{1}{2}}+9}{a-9}\end{array}$

Hence the simplified form of the expression is $\frac{a-6a^{\frac{1}{2}}+9}{a-9}$ .