Chapter R - Section R.7 - Radical Expressions - R.7 Exercises - Page 68: 58

$\dfrac{gh^2\sqrt{ghr}}{r^2}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To simplify the given expression, $\sqrt{\dfrac{g^3h^5}{r^3}} ,$ make the denominator a perfect power of the index so that the final result will already be in rationalized form. Then find a factor of the radicand that is a perfect power of the index. Finally, extract the root of that factor. $\bf{\text{Solution Details:}}$ Multiplying the radicand that will make the denominator a perfect power of the index results to \begin{array}{l}\require{cancel} \sqrt{\dfrac{g^3h^5}{r^3}\cdot\dfrac{r}{r}} \\\\= \sqrt{\dfrac{g^3h^5r}{r^4}} .\end{array} Factoring the radicand into an expression that is a perfect power of the index and then extracting its root result to \begin{array}{l}\require{cancel} \sqrt{\dfrac{g^2h^4}{r^4}\cdot ghr} \\\\= \sqrt{\left( \dfrac{gh^2}{r^2} \right)^2\cdot ghr} \\\\= \left| \dfrac{gh^2}{r^2} \right|\sqrt{ghr} .\end{array} Since all variables are assumed to be positive, the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{gh^2}{r^2}\sqrt{ghr} \\\\= \dfrac{gh^2\sqrt{ghr}}{r^2} .\end{array}

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