# Chapter 7 - Section 7.2 - Rational Exponents - 7.2 Exercises - Page 449: 92

$y^{3/2}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ Use the definition of rational exponents and the laws of exponents to simplify the given expression, $\sqrt[6]{y^5}\cdot\sqrt[3]{y^2} .$ $\bf{\text{Solution Details:}}$ 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} y^{5/6}\cdot y^{2/3} .\end{array} Using the Product Rule of the laws of exponents which is given by $x^m\cdot x^n=x^{m+n},$ the expression above is equivalent to \begin{array}{l}\require{cancel} y^{\frac{5}{6}+\frac{2}{3}} .\end{array} To simplify the expression $\dfrac{5}{6}+\dfrac{2}{3} ,$ change the expressions to similar fractions (same denominator) by using the $LCD$. The $LCD$ of the denominators $6$ and $3$ is $6$ since it is the lowest number that can be exactly divided by the denominators. Multiplying the terms by an expression equal to $1$ that will make the denominator equal to the $LCD$ results to \begin{array}{l}\require{cancel} y^{\frac{5}{6}+\frac{2}{3}\cdot\frac{2}{2}} \\\\= y^{\frac{5}{6}+\frac{4}{6}} \\\\= y^{\frac{9}{6}} \\\\= y^{\frac{3}{2}} \\\\= y^{3/2} .\end{array}

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