# Chapter 7 - Section 7.2 - Rational Exponents - 7.2 Exercises: 100

$x^{1/16}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ Use the definition of rational exponents and the laws of exponents to simplify the given expression, $\sqrt{\sqrt{\sqrt{\sqrt{x}}}} .$ $\bf{\text{Solution Details:}}$ Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \sqrt[2]{\sqrt[2]{\sqrt[2]{\sqrt[2]{x}}}} \\\\= \sqrt[2]{\sqrt[2]{\sqrt[2]{x^{1/2}}}} \\\\= \sqrt[2]{\sqrt[2]{(x^{1/2})^{1/2}}} \\\\= \sqrt[2]{((x^{1/2})^{1/2})^{1/2}} \\\\= (((x^{1/2})^{1/2})^{1/2})^{1/2} .\end{array} Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the expression above is equivalent to \begin{array}{l}\require{cancel} x^{\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}} \\\\= x^{\frac{1}{16}} \\\\= x^{1/16} .\end{array}

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