Exercises: Terms with Square Roots

Simplify the following terms.

$\sqrt{500}+3\sqrt{98}-5\sqrt8-3\sqrt{45}$

$\sqrt{64k^2}$

$\left(\sqrt{\frac{x^5y}{5a}}:\sqrt{\frac{x^3y^3}{a^2}}\right)\cdot\sqrt{\frac{25x}a}\;\;\;\;\;\;\;\;\left(x,\;y,\;z\;>\;0\right)$

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	$=$	$\displaystyle \sqrt{500}+3\sqrt{98}-5\sqrt{8}-3\sqrt{45}$
	↓	Decompose the numbers under the roots.
	$=$	$\displaystyle \sqrt{5\cdot10^2}+3\sqrt{2\cdot7^2}-5\sqrt{2^2\cdot2}-3\sqrt{3^2\cdot5}$
	↓	Pull the powers out of the root.
	$=$	$\displaystyle 10\sqrt{5}+3\cdot7\sqrt{2}-5\cdot2\sqrt{2}-3\cdot3\sqrt{5}$
	$=$	$\displaystyle 10\sqrt{5}+21\sqrt{2}-10\sqrt{2}-9\sqrt{5}$
	$=$	$\displaystyle \sqrt{5}+11\sqrt{2}$

	$=$	$\displaystyle \sqrt{64k^2}$
	↓	Decompose the numbers under the root.
	$=$	$\displaystyle \sqrt{8^2\cdot k^2}$
	↓	Pull the powers out of the root. (Don't forget the absolute value around k.)
	$=$	$\displaystyle 8\cdot\left\|k\right\|$

	$=$	$\displaystyle \left(\sqrt{\frac{x^5y}{5a}}:\sqrt{\frac{x^3y^3}{a^2}}\right)\cdot\sqrt{\frac{25x}{a}}$
	↓	Conclude the fractions.
	$=$	$\displaystyle \frac{\sqrt{x^5y}\cdot\sqrt{a^2}}{\sqrt{5a}\cdot\sqrt{x^3y^3}}\cdot\frac{\sqrt{25x}}{\sqrt{a}}$
	$=$	$\displaystyle \sqrt{\frac{x^5y\cdot a^2\cdot25x}{5ax^3y^3a}}$
	↓	Shorten the fraction.
	$=$	$\displaystyle \sqrt{\frac{5x^3}{y^2}}$
	↓	Pull the powers out of the root.
	$=$	$\displaystyle \frac{x}{y}\sqrt{5x}$

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