Chapter A.10, Problem 98AYU

To determine

To dtermine: Express the expression $\frac{{\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-x^2{\left(x^2+4\right)}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{x^2+4}$ as a single quotient in which only positive exponent or radical appear.

Answer to Problem 98AYU

The required form of the given expression $\frac{{\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-x^2{\left(x^2+4\right)}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{x^2+4}$ is $\frac{4}{{\left(x^2+4\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$.

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{{\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-x^2{\left(x^2+4\right)}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{x^2+4}$,

$\frac{{\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-x^2{\left(x^2+4\right)}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{x^2+4}=\frac{{\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-\frac{x^2}{{\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{x^2+4}$

Simplify the numerator expression.

Now consider ${\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-\frac{x^2}{{\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$,

${\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-\frac{x^2}{{\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}=\frac{\left(x^2+4\right)-x^2}{{\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}= \frac{4}{{\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

$\frac{{\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-x^2{\left(x^2+4\right)}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{x^2+4}=\frac{\frac{4}{{\left(x^2+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{x^2+4}=\frac{4}{{\left(x^2+4\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$