Dividing square roots can often be simplified by combining the radicals into a single square root using \(\frac{\sqrt{a}}{\sqrt{m}}=\sqrt{\frac{a}{m}}\). After combining, simplify the fraction and square root whenever possible. When a radical remains in the denominator, the denominator can be rationalized by multiplying the numerator and denominator by an appropriate radical.
Notes
Dividing Square Roots
\(\displaystyle\frac{\sqrt{a}}{\sqrt{m}}= \)\(\sqrt{\frac{a}{m}}\)
Practice Problems
\(\textbf{1)}\) \(\displaystyle\frac{\sqrt{45}}{\sqrt{5}}\)
The answer is \(3\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{45}}{\sqrt{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{45}{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{9}\)
\(\,\,\,\,\,3\)
The answer is \(3\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{45}}{\sqrt{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{45}{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{9}\)
\(\,\,\,\,\,3\)
The answer is \(3\)
\(\textbf{2)}\) \(\displaystyle\frac{\sqrt{12}}{\sqrt{3}}\)
The answer is \(2\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{12}}{\sqrt{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{12}{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{4}\)
\(\,\,\,\,\,2\)
The answer is \(2\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{12}}{\sqrt{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{12}{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{4}\)
\(\,\,\,\,\,2\)
The answer is \(2\)
\(\textbf{3)}\) \(\displaystyle\frac{\sqrt{32}}{\sqrt{2}}\)
The answer is \(4\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{32}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{32}{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{16}\)
\(\,\,\,\,\,4\)
The answer is \(4\)
![Link to Youtube Video Solving Question Number 3](https://andymath.com/wp-content/uploads/2020/08/play-video.jpg")
\(\,\,\,\,\,\displaystyle\frac{\sqrt{32}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{32}{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{16}\)
\(\,\,\,\,\,4\)
The answer is \(4\)
\(\textbf{4)}\) \(\displaystyle\frac{\sqrt{8}}{\sqrt{2}}\)
The answer is \(2\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{8}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{8}{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{4}\)
\(\,\,\,\,\,2\)
The answer is \(2\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{8}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{8}{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{4}\)
\(\,\,\,\,\,2\)
The answer is \(2\)
\(\textbf{5)}\) \(\displaystyle\frac{\sqrt{27}}{\sqrt{3}}\)
The answer is \(3\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{27}}{\sqrt{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{27}{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{9}\)
\(\,\,\,\,\,3\)
The answer is \(3\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{27}}{\sqrt{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{27}{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{9}\)
\(\,\,\,\,\,3\)
The answer is \(3\)
\(\textbf{6)}\) \(\displaystyle\frac{\sqrt{75}}{\sqrt{3}}\)
The answer is \(5\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{75}}{\sqrt{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{75}{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{25}\)
\(\,\,\,\,\,5\)
The answer is \(5\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{75}}{\sqrt{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{75}{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{25}\)
\(\,\,\,\,\,5\)
The answer is \(5\)
\(\textbf{7)}\) \(\displaystyle\frac{\sqrt{72}}{\sqrt{2}}\)
The answer is \(6\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{72}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{72}{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{36}\)
\(\,\,\,\,\,6\)
The answer is \(6\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{72}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{72}{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{36}\)
\(\,\,\,\,\,6\)
The answer is \(6\)
\(\textbf{8)}\) \(\displaystyle\frac{\sqrt{5}}{\sqrt{5}}\)
The answer is \(1\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{5}}{\sqrt{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{5}{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{1}\)
\(\,\,\,\,\,1\)
The answer is \(1\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{5}}{\sqrt{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{5}{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{1}\)
\(\,\,\,\,\,1\)
The answer is \(1\)
\(\textbf{9)}\) \(\displaystyle\frac{\sqrt{18}}{\sqrt{2}}\)
The answer is \(3\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{18}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{18}{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{9}\)
\(\,\,\,\,\,3\)
The answer is \(3\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{18}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{18}{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{9}\)
\(\,\,\,\,\,3\)
The answer is \(3\)
\(\textbf{10)}\) \(\displaystyle\frac{\sqrt{300}}{\sqrt{3}}\)
The answer is \(10\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{300}}{\sqrt{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{300}{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{100}\)
\(\,\,\,\,\,10\)
The answer is \(10\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{300}}{\sqrt{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{300}{3}}\)
\(\,\,\,\,\,\displaystyle\sqrt{100}\)
\(\,\,\,\,\,10\)
The answer is \(10\)
\(\textbf{11)}\) \(\displaystyle\frac{\sqrt{50}}{\sqrt{2}}\)
The answer is \(5\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{50}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{50}{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{25}\)
\(\,\,\,\,\,5\)
The answer is \(5\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{50}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{50}{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{25}\)
\(\,\,\,\,\,5\)
The answer is \(5\)
\(\textbf{12)}\) \(\displaystyle\frac{\sqrt{18}}{\sqrt{8}}\)
The answer is \(\frac{3}{2}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{18}}{\sqrt{8}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{18}{8}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{9}{4}}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{9}}{\sqrt{4}}\)
\(\,\,\,\,\,\displaystyle\frac{3}{2}\)
The answer is \(\frac{3}{2}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{18}}{\sqrt{8}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{18}{8}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{9}{4}}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{9}}{\sqrt{4}}\)
\(\,\,\,\,\,\displaystyle\frac{3}{2}\)
The answer is \(\frac{3}{2}\)
\(\textbf{13)}\) \(\displaystyle\frac{\sqrt{12}}{\sqrt{27}}\)
The answer is \(\frac{2}{3}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{12}}{\sqrt{27}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{12}{27}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{4}{9}}\)
\(\,\,\,\,\,\displaystyle\frac{2}{3}\)
The answer is \(\frac{2}{3}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{12}}{\sqrt{27}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{12}{27}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{4}{9}}\)
\(\,\,\,\,\,\displaystyle\frac{2}{3}\)
The answer is \(\frac{2}{3}\)
\(\textbf{14)}\) \(\displaystyle\frac{\sqrt{98}}{\sqrt{2}}\)
The answer is \(7\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{98}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{98}{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{49}\)
\(\,\,\,\,\,7\)
The answer is \(7\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{98}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{98}{2}}\)
\(\,\,\,\,\,\displaystyle\sqrt{49}\)
\(\,\,\,\,\,7\)
The answer is \(7\)
\(\textbf{15)}\) \(\displaystyle\frac{\sqrt{20}}{\sqrt{5}}\)
The answer is \(2\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{20}}{\sqrt{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{20}{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{4}\)
\(\,\,\,\,\,2\)
The answer is \(2\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{20}}{\sqrt{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{20}{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{4}\)
\(\,\,\,\,\,2\)
The answer is \(2\)
\(\textbf{16)}\) \(\displaystyle\frac{\sqrt{10}}{\sqrt{5}}\)
The answer is \(\sqrt{2}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{10}}{\sqrt{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{10}{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{2}\)
The answer is \(\sqrt{2}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{10}}{\sqrt{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{\frac{10}{5}}\)
\(\,\,\,\,\,\displaystyle\sqrt{2}\)
The answer is \(\sqrt{2}\)
\(\textbf{17)}\) \(\displaystyle\frac{\sqrt{5}}{\sqrt{2}}\)
The answer is \(\frac{\sqrt{10}}{2}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{5}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{5}}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{10}}{\sqrt{4}}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{10}}{2}\)
The answer is \(\frac{\sqrt{10}}{2}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{5}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{5}}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{10}}{\sqrt{4}}\)
\(\,\,\,\,\,\displaystyle\frac{\sqrt{10}}{2}\)
The answer is \(\frac{\sqrt{10}}{2}\)
\(\textbf{18)}\) \(\displaystyle\frac{3}{\sqrt{5}}\)
The answer is \(\frac{3\sqrt{5}}{5}\)
\(\,\,\,\,\,\displaystyle\frac{3}{\sqrt{5}}\)
\(\,\,\,\,\,\displaystyle\frac{3}{\sqrt{5}}\cdot\frac{\sqrt{5}}{\sqrt{5}}\)
\(\,\,\,\,\,\displaystyle\frac{3\sqrt{5}}{5}\)
The answer is \(\frac{3\sqrt{5}}{5}\)
\(\,\,\,\,\,\displaystyle\frac{3}{\sqrt{5}}\)
\(\,\,\,\,\,\displaystyle\frac{3}{\sqrt{5}}\cdot\frac{\sqrt{5}}{\sqrt{5}}\)
\(\,\,\,\,\,\displaystyle\frac{3\sqrt{5}}{5}\)
The answer is \(\frac{3\sqrt{5}}{5}\)
\(\textbf{19)}\) \(\displaystyle\frac{4\sqrt{27}}{2\sqrt{3}}\)
The answer is \(6\)
\(\,\,\,\,\,\displaystyle\frac{4\sqrt{27}}{2\sqrt{3}}\)
\(\,\,\,\,\,\displaystyle2\cdot\frac{\sqrt{27}}{\sqrt{3}}\)
\(\,\,\,\,\,\displaystyle2\sqrt{\frac{27}{3}}\)
\(\,\,\,\,\,2\sqrt{9}\)
\(\,\,\,\,\,2(3)\)
\(\,\,\,\,\,6\)
The answer is \(6\)
\(\,\,\,\,\,\displaystyle\frac{4\sqrt{27}}{2\sqrt{3}}\)
\(\,\,\,\,\,\displaystyle2\cdot\frac{\sqrt{27}}{\sqrt{3}}\)
\(\,\,\,\,\,\displaystyle2\sqrt{\frac{27}{3}}\)
\(\,\,\,\,\,2\sqrt{9}\)
\(\,\,\,\,\,2(3)\)
\(\,\,\,\,\,6\)
The answer is \(6\)
\(\textbf{20)}\) \(\displaystyle\frac{2\sqrt{6}}{\sqrt{3}}\)
The answer is \(2\sqrt{2}\)
\(\,\,\,\,\,\displaystyle\frac{2\sqrt{6}}{\sqrt{3}}\)
\(\,\,\,\,\,2\sqrt{\frac{6}{3}}\)
\(\,\,\,\,\,2\sqrt{2}\)
The answer is \(2\sqrt{2}\)
\(\,\,\,\,\,\displaystyle\frac{2\sqrt{6}}{\sqrt{3}}\)
\(\,\,\,\,\,2\sqrt{\frac{6}{3}}\)
\(\,\,\,\,\,2\sqrt{2}\)
The answer is \(2\sqrt{2}\)
See Related Pages\(\)
\(\bullet\text{ Simplifying Square Roots}\)
\(\,\,\,\,\,\,\,\,\sqrt{32}=4\sqrt{2}…\)
\(\bullet\text{ Multiplying Square Roots}\)
\(\,\,\,\,\,\,\,\, \sqrt{5} \cdot \sqrt{125} …\)
\(\bullet\text{ Adding and Subtracting Square Roots}\)
\(\,\,\,\,\,\,\,\,3\sqrt{2}-7\sqrt{2}…\)
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