Chapter 3.3, Problem 2QR

To determine

The expression by writing it as a sum of powers of x .

Answer to Problem 2QR

The required expression is $x+\frac{1}{x}$ .

Explanation of Solution

Given information:

The expression $={\left(\frac{x}{x^2+1}\right)}^{-1}$

Formula used:

Reciprocal of a fraction:

  ${\left(\frac{x}{y}\right)}^{-1}=\frac{y}{x}$

Calculation:

The given expression is ${\left(\frac{x}{x^2+1}\right)}^{-1}$ .

To write the above expression as a sum of powers of x , use the reciprocal of a fraction.

  $\begin{array}{c}{\left(\frac{x}{x^2+1}\right)}^{-1}=\frac{x^2+1}{x}\\ =\frac{x^2}{x}+\frac{1}{x}\\ =x+\frac{1}{x}\end{array}$

Hence, the required expression is $x+\frac{1}{x}$ .