Chapter 7 - Section 7.4 - Adding, Subtracting, and Dividing Radical Expressions - Exercise Set - Page 539: 57

$3\sqrt{xy}$

Work Step by Step

The given expression can be written as: $=\dfrac{1}{2}\cdot \dfrac{\sqrt{72xy}}{\sqrt{2}}$ RECALL: (1) The quotient rule: $\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\dfrac{a}{b}}$ where $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers and $b\ne0$ (2) $\dfrac{a^m}{a^n} = a^{m-n}, a \ne =0$ Use the quotient rule above to obtain: $\require{cancel}=\dfrac{1}{2} \cdot \sqrt{\dfrac{72xy}{2}} \\=\dfrac{1}{2} \cdot \sqrt{\dfrac{36\cancel{72}xy}{\cancel{2}}} \\=\dfrac{1}{2} \cdot \sqrt{36xy}$ Factor the radicand so that at least one factor is a perfect square to obtain: $=\dfrac{1}{2} \cdot \sqrt{36(xy)} \\=\dfrac{1}{2} \cdot \sqrt{6^2(xy)}$ Simplify to obtain: $=\frac{1}{2} \cdot 6\sqrt{xy} \\=3\sqrt{xy}$

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