# Chapter 10 - Exponents and Radicals - Mid-Chapter Review - Mixed Review - Page 663: 17

$\sqrt[8]{t}$

#### Work Step by Step

Using $x^{m/n}=\sqrt[n]{x^m}=\left(\sqrt[n]{x} \right)^m,$ then \begin{array}{l}\require{cancel} \dfrac{\sqrt{t}}{\sqrt[8]{t^3}} \\\\= \dfrac{t^{1/2}}{t^{3/8}} .\end{array} Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to \begin{array}{l}\require{cancel} \dfrac{t^{1/2}}{t^{3/8}} \\\\= t^{\frac{1}{2}-\frac{3}{8}} \\\\= t^{\frac{4}{8}-\frac{3}{8}} \\\\= t^{\frac{1}{8}} .\end{array} Using $x^{m/n}=\sqrt[n]{x^m}=\left(\sqrt[n]{x} \right)^m,$ then \begin{array}{l}\require{cancel} t^{\frac{1}{8}} \\\\= \sqrt[8]{t^1} \\\\= \sqrt[8]{t} .\end{array}

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