Chapter 6.6, Problem 69HP

To determine

Explain how it might be easier to simplify an expression using rational exponents rather than using radicals.

Explanation of Solution

Formula Used:

  $\sqrt[n]{x^m}=x^{\frac{m}{n}}={\left(\sqrt[n]{x}\right)}^m$

  $x^{m+n}=x^m\cdot x^n$

  $x^{m-n}=\frac{x^m}{x^n}$

Product property of radicals : $\sqrt[x]{a}\cdot \sqrt[x]{b}=\sqrt[x]{ab}$

Quotient property of radicals : $\frac{\sqrt[x]{a}}{\sqrt[x]{b}}=\sqrt[x]{\frac{a}{b}}$

Calculation:

Consider:

  $\frac{\sqrt[4]{x^6{y}^2}}{\sqrt[2]{x}\cdot \sqrt[8]{x^4{y}^{16}}}$

If we work with the radicals , we need the index of the radicals equal so as to apply the Product property of radicals or Quotient property of radicals, hence , it becomes a bit difficult here , since we have to convert the radicals with index 2 and 4 into the radicals of index 8 first, and then simplify.

But , if we convert the radicals into the rational exponents using $\sqrt[n]{x^m}=x^{\frac{m}{n}}$ , the simplification process becomes much more easy:

  $\begin{array}{l}\frac{\sqrt[4]{x^6{y}^4}}{\sqrt[2]{x}\cdot \sqrt[8]{x^4{y}^4}}\\ =\frac{{\left(x^6{y}^4\right)}^{\frac{1}{4}}}{x^{\frac{1}{2}}\cdot {\left(x^4{y}^4\right)}^{\frac{1}{8}}}\mathrm{.............}[\sqrt[n]{x^m}=x^{\frac{m}{n}}]\\ =\frac{x^{\frac{3}{2}}\cdot y}{x^{\frac{1}{2}}\cdot x^{\frac{1}{2}}\cdot y^{\frac{1}{2}}}\mathrm{.......................}[{\left(x^m\right)}^n=x^{mn}]\\ =\frac{x^{\frac{3}{2}}\cdot y}{x^{\frac{1}{2}+\frac{1}{2}}\cdot y^{\frac{1}{2}}}\mathrm{..........................}[x^{m+n}=x^m\cdot x^n]\\ =\frac{x^{\frac{3}{2}}\cdot y}{x\cdot y^{\frac{1}{2}}}\\ =x^{\frac{3}{2}-1}\cdot y^{1-\frac{1}{2}}\mathrm{......................}[x^{m-n}=\frac{x^m}{x^n}]\\ =x^{\frac{1}{2}}{y}^{\frac{1}{2}}\\ =\sqrt{x}\cdot \sqrt{y}\\ =\sqrt{xy}\end{array}$