Chapter A.10, Problem 99AYU

To determine

Express the expression $\frac{\frac{1+x^2}{2\sqrt{x}}-2x\sqrt{x}}{{\left(1+x^2\right)}^2}$ as a single quotient in which only positive exponent or radical appear.

Answer to Problem 99AYU

The required form of the given expression $\frac{\frac{1+x^2}{2\sqrt{x}}-2x\sqrt{x}}{{\left(1+x^2\right)}^2}$ is $\frac{1-3x^2}{2\sqrt{x}{\left(1+x^2\right)}^2}$.

Explanation of Solution

Given:

It is asked to determine the expression as a single quotient in which only positive exponents or radicals appear.

Calculation:

Consider $\frac{\frac{1+x^2}{2\sqrt{x}}-2x\sqrt{x}}{{\left(1+x^2\right)}^2}$,

Simplify the numerator expression.

Now consider $\frac{1+x^2}{2\sqrt{x}}-2x\sqrt{x}$,

$\frac{1+x^2}{2\sqrt{x}}-2x\sqrt{x}=\frac{1+x^2-2\sqrt{x}\times 2x\sqrt{x}}{2\sqrt{x}}= \frac{1-3x^2}{2\sqrt{x}}$

$\frac{\frac{1+x^2}{2\sqrt{x}}-2x\sqrt{x}}{{\left(1+x^2\right)}^2}=\frac{\frac{1-3x^2}{2\sqrt{x}}}{{\left(1+x^2\right)}^2}=\frac{1-3x^2}{2\sqrt{x}{\left(1+x^2\right)}^2}$