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Quiz 24
1.) Simplify $\sqrt{100x^2}$ (assuming $x \geq 0$).
Solution: Using the rule that $\sqrt{A} \sqrt{B} = \sqrt{A \cdot B}$, calculate $$\sqrt{100x^2} = \sqrt{100} \sqrt{x^2} = 10x.$$

2.) Write as one square root: $$\left( \sqrt{3} \right) \left( \sqrt{7} \right).$$ Solution: Using the rule $\sqrt{A} \sqrt{B} = \sqrt{A \cdot B}$, calculate $$(\sqrt{3})(\sqrt{7}) = \sqrt{3 \cdot 7} = \sqrt{21}.$$

3.) Simplify $$\sqrt{ \dfrac{36y^2}{4x^4}}.$$ Solution: Using the rules $\sqrt{A}\sqrt{B}=\sqrt{A \cdot B}$ and $\dfrac{\sqrt{A}}{\sqrt{B}}=\sqrt{\dfrac{A}{B}}$, calculate $$\begin{array}{ll} \sqrt{\dfrac{36}y^2}{4x^2} &= \dfrac{\sqrt{36y^2}}{\sqrt{4x^4}} \\ &= \dfrac{\sqrt{36}\sqrt{y^2}}{\sqrt{4}\sqrt{x^4}} \\ &= \dfrac{6y}{2x^2}. \end{array}$$

4.) Simplify (assuming $x \geq 0$) $$\sqrt{x^{15}}.$$ Solution: Calculate $$\sqrt{x^{15}}=\sqrt{x \cdot x^{14}} = \sqrt{x} \sqrt{x^{14}} = \sqrt{x} \sqrt{\left( x^7 \right)^2}= x^7 \sqrt{x}.$$

5.) Simplify (assuming $x \geq 0$) $$\sqrt{ \dfrac{48x^3}{12x^5}}.$$ Solution: Calculate $$\begin{array}{ll} \sqrt{ \dfrac{48x^3}{12x^5} } &= \dfrac{\sqrt{48}\sqrt{x^3}}{\sqrt{12}\sqrt{x^5}} \\ &= \dfrac{\sqrt{4^2 \cdot 3} \sqrt{x} \sqrt{x^2}}{\sqrt{2^2 \cdot 3} \sqrt{x} \sqrt{x^4}} \\ &=\dfrac{4\sqrt{3}\sqrt{x} x}{2\sqrt{3}\sqrt{x} x^2} \\ &=\dfrac{4x}{2x^2} \\ &= \dfrac{2}{x}. \end{array}$$