Simplifying square roots means finding perfect-square factors inside a radical and moving their square roots outside. For example, \(\sqrt{48}=\sqrt{16\cdot3}=4\sqrt{3}\). The goal is to rewrite each radical so that no perfect-square factor greater than 1 remains inside the square root.
Lesson
Practice Problems
Simplify
\(\textbf{1)}\) \( \sqrt{24} \) The answer is \( 2\sqrt{6} \)
\(\,\,\,\,\,\,\sqrt{24}\)
\(\,\,\,\,\,\,\sqrt{4\cdot6}\)
\(\,\,\,\,\,\,\sqrt{4}\cdot\sqrt{6}\)
\(\,\,\,\,\,\,2\sqrt{6}\)
The answer is \(2\sqrt{6}\)
\(\,\,\,\,\,\,\sqrt{24}\)
\(\,\,\,\,\,\,\sqrt{4\cdot6}\)
\(\,\,\,\,\,\,\sqrt{4}\cdot\sqrt{6}\)
\(\,\,\,\,\,\,2\sqrt{6}\)
The answer is \(2\sqrt{6}\)
\(\textbf{2)}\) \( \sqrt{48} \) The answer is \( 4\sqrt{3} \)
\(\,\,\,\,\,\,\sqrt{48}\)
\(\,\,\,\,\,\,\sqrt{16\cdot3}\)
\(\,\,\,\,\,\,\sqrt{16}\cdot\sqrt{3}\)
\(\,\,\,\,\,\,4\sqrt{3}\)
The answer is \(4\sqrt{3}\)
\(\,\,\,\,\,\,\sqrt{48}\)
\(\,\,\,\,\,\,\sqrt{16\cdot3}\)
\(\,\,\,\,\,\,\sqrt{16}\cdot\sqrt{3}\)
\(\,\,\,\,\,\,4\sqrt{3}\)
The answer is \(4\sqrt{3}\)
\(\textbf{3)}\) \( \sqrt{120} \) The answer is \( 2\sqrt{30} \)
\(\,\,\,\,\,\,\sqrt{120}\)
\(\,\,\,\,\,\,\sqrt{4\cdot30}\)
\(\,\,\,\,\,\,\sqrt{4}\cdot\sqrt{30}\)
\(\,\,\,\,\,\,2\sqrt{30}\)
The answer is \(2\sqrt{30}\)
\(\,\,\,\,\,\,\sqrt{120}\)
\(\,\,\,\,\,\,\sqrt{4\cdot30}\)
\(\,\,\,\,\,\,\sqrt{4}\cdot\sqrt{30}\)
\(\,\,\,\,\,\,2\sqrt{30}\)
The answer is \(2\sqrt{30}\)
\(\textbf{4)}\) \( \sqrt{32} \) The answer is \( 4\sqrt{2} \)
\(\,\,\,\,\,\, \sqrt{32} \)
\(\,\,\,\,\,\, \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2} \)
\(\,\,\,\,\,\, \sqrt{(2 \cdot 2) \cdot (2 \cdot 2) \cdot 2} \)
\(\,\,\,\,\,\, \sqrt{4 \cdot 4 \cdot 2} \)
\(\,\,\,\,\,\, \sqrt{4} \cdot \sqrt{4} \cdot \sqrt{2} \)
\(\,\,\,\,\,\, 2 \cdot 2 \cdot \sqrt{2} \)
\(\,\,\,\,\,\,4\sqrt{2}\)
The answer is \(4\sqrt{2}\)
\(\,\,\,\,\,\, \sqrt{32} \)
\(\,\,\,\,\,\, \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2} \)
\(\,\,\,\,\,\, \sqrt{(2 \cdot 2) \cdot (2 \cdot 2) \cdot 2} \)
\(\,\,\,\,\,\, \sqrt{4 \cdot 4 \cdot 2} \)
\(\,\,\,\,\,\, \sqrt{4} \cdot \sqrt{4} \cdot \sqrt{2} \)
\(\,\,\,\,\,\, 2 \cdot 2 \cdot \sqrt{2} \)
\(\,\,\,\,\,\,4\sqrt{2}\)
The answer is \(4\sqrt{2}\)
\(\textbf{5)}\) \( \sqrt{180} \) The answer is \( 6\sqrt{5} \)
\(\,\,\,\,\,\, \sqrt{180} \)
\(\,\,\,\,\,\, \sqrt{2 \cdot 2 \cdot 3 \cdot 3 \cdot 5} \)
\(\,\,\,\,\,\, \sqrt{(2 \cdot 2) \cdot (3 \cdot 3) \cdot 5} \)
\(\,\,\,\,\,\, \sqrt{4 \cdot 9 \cdot 5} \)
\(\,\,\,\,\,\, \sqrt{4} \cdot \sqrt{9} \cdot \sqrt{5} \)
\(\,\,\,\,\,\, 2 \cdot 3 \cdot \sqrt{5} \)
\(\,\,\,\,\,\, 6\sqrt{5} \)
The answer is \(6\sqrt{5}\)
\(\,\,\,\,\,\, \sqrt{180} \)
\(\,\,\,\,\,\, \sqrt{2 \cdot 2 \cdot 3 \cdot 3 \cdot 5} \)
\(\,\,\,\,\,\, \sqrt{(2 \cdot 2) \cdot (3 \cdot 3) \cdot 5} \)
\(\,\,\,\,\,\, \sqrt{4 \cdot 9 \cdot 5} \)
\(\,\,\,\,\,\, \sqrt{4} \cdot \sqrt{9} \cdot \sqrt{5} \)
\(\,\,\,\,\,\, 2 \cdot 3 \cdot \sqrt{5} \)
\(\,\,\,\,\,\, 6\sqrt{5} \)
The answer is \(6\sqrt{5}\)
\(\textbf{6)}\) \( \sqrt{360} \) The answer is \( 6\sqrt{10} \)
\(\,\,\,\,\,\,\sqrt{360}\)
\(\,\,\,\,\,\,\sqrt{36\cdot10}\)
\(\,\,\,\,\,\,\sqrt{36}\cdot\sqrt{10}\)
\(\,\,\,\,\,\,6\sqrt{10}\)
The answer is \(6\sqrt{10}\)
\(\,\,\,\,\,\,\sqrt{360}\)
\(\,\,\,\,\,\,\sqrt{36\cdot10}\)
\(\,\,\,\,\,\,\sqrt{36}\cdot\sqrt{10}\)
\(\,\,\,\,\,\,6\sqrt{10}\)
The answer is \(6\sqrt{10}\)
\(\textbf{7)}\) \( \sqrt{75} \) The answer is \( 5\sqrt{3} \)
\(\,\,\,\,\,\,\sqrt{75}\)
\(\,\,\,\,\,\,\sqrt{25\cdot3}\)
\(\,\,\,\,\,\,\sqrt{25}\cdot\sqrt{3}\)
\(\,\,\,\,\,\,5\sqrt{3}\)
The answer is \(5\sqrt{3}\)
![Link to Youtube Video Solving Question Number 7](https://andymath.com/wp-content/uploads/2020/08/play-video.jpg")
\(\,\,\,\,\,\,\sqrt{75}\)
\(\,\,\,\,\,\,\sqrt{25\cdot3}\)
\(\,\,\,\,\,\,\sqrt{25}\cdot\sqrt{3}\)
\(\,\,\,\,\,\,5\sqrt{3}\)
The answer is \(5\sqrt{3}\)
\(\textbf{8)}\) \( \sqrt{240} \) The answer is \( 4\sqrt{15} \)
\(\,\,\,\,\,\,\sqrt{240}\)
\(\,\,\,\,\,\,\sqrt{16\cdot15}\)
\(\,\,\,\,\,\,\sqrt{16}\cdot\sqrt{15}\)
\(\,\,\,\,\,\,4\sqrt{15}\)
The answer is \(4\sqrt{15}\)
\(\,\,\,\,\,\,\sqrt{240}\)
\(\,\,\,\,\,\,\sqrt{16\cdot15}\)
\(\,\,\,\,\,\,\sqrt{16}\cdot\sqrt{15}\)
\(\,\,\,\,\,\,4\sqrt{15}\)
The answer is \(4\sqrt{15}\)
\(\textbf{9)}\) \( \sqrt{98} \) The answer is \( 7\sqrt{2} \)
\(\,\,\,\,\,\,\sqrt{98}\)
\(\,\,\,\,\,\,\sqrt{49\cdot2}\)
\(\,\,\,\,\,\,\sqrt{49}\cdot\sqrt{2}\)
\(\,\,\,\,\,\,7\sqrt{2}\)
The answer is \(7\sqrt{2}\)
\(\,\,\,\,\,\,\sqrt{98}\)
\(\,\,\,\,\,\,\sqrt{49\cdot2}\)
\(\,\,\,\,\,\,\sqrt{49}\cdot\sqrt{2}\)
\(\,\,\,\,\,\,7\sqrt{2}\)
The answer is \(7\sqrt{2}\)
\(\textbf{10)}\) \( \sqrt{54} \) The answer is \( 3\sqrt{6} \)
\(\,\,\,\,\,\,\sqrt{54}\)
\(\,\,\,\,\,\,\sqrt{9\cdot6}\)
\(\,\,\,\,\,\,\sqrt{9}\cdot\sqrt{6}\)
\(\,\,\,\,\,\,3\sqrt{6}\)
The answer is \(3\sqrt{6}\)
\(\,\,\,\,\,\,\sqrt{54}\)
\(\,\,\,\,\,\,\sqrt{9\cdot6}\)
\(\,\,\,\,\,\,\sqrt{9}\cdot\sqrt{6}\)
\(\,\,\,\,\,\,3\sqrt{6}\)
The answer is \(3\sqrt{6}\)
\(\textbf{11)}\) \(\sqrt{72}\) The answer is \(6\sqrt{2}\)
\(\,\,\,\,\,\,\sqrt{72}\)
\(\,\,\,\,\,\,\sqrt{36\cdot2}\)
\(\,\,\,\,\,\,\sqrt{36}\cdot\sqrt{2}\)
\(\,\,\,\,\,\,6\sqrt{2}\)
The answer is \(6\sqrt{2}\)
\(\,\,\,\,\,\,\sqrt{72}\)
\(\,\,\,\,\,\,\sqrt{36\cdot2}\)
\(\,\,\,\,\,\,\sqrt{36}\cdot\sqrt{2}\)
\(\,\,\,\,\,\,6\sqrt{2}\)
The answer is \(6\sqrt{2}\)
\(\textbf{12)}\) \(\sqrt{200}\) The answer is \(10\sqrt{2}\)
\(\,\,\,\,\,\,\sqrt{200}\)
\(\,\,\,\,\,\,\sqrt{100\cdot2}\)
\(\,\,\,\,\,\,\sqrt{100}\cdot\sqrt{2}\)
\(\,\,\,\,\,\,10\sqrt{2}\)
The answer is \(10\sqrt{2}\)
\(\,\,\,\,\,\,\sqrt{200}\)
\(\,\,\,\,\,\,\sqrt{100\cdot2}\)
\(\,\,\,\,\,\,\sqrt{100}\cdot\sqrt{2}\)
\(\,\,\,\,\,\,10\sqrt{2}\)
The answer is \(10\sqrt{2}\)
\(\textbf{13)}\) \(\sqrt{45}\) The answer is \(3\sqrt{5}\)
\(\,\,\,\,\,\,\sqrt{45}\)
\(\,\,\,\,\,\,\sqrt{9\cdot5}\)
\(\,\,\,\,\,\,\sqrt{9}\cdot\sqrt{5}\)
\(\,\,\,\,\,\,3\sqrt{5}\)
The answer is \(3\sqrt{5}\)
\(\,\,\,\,\,\,\sqrt{45}\)
\(\,\,\,\,\,\,\sqrt{9\cdot5}\)
\(\,\,\,\,\,\,\sqrt{9}\cdot\sqrt{5}\)
\(\,\,\,\,\,\,3\sqrt{5}\)
The answer is \(3\sqrt{5}\)
\(\textbf{14)}\) \(\sqrt{147}\) The answer is \(7\sqrt{3}\)
\(\,\,\,\,\,\,\sqrt{147}\)
\(\,\,\,\,\,\,\sqrt{49\cdot3}\)
\(\,\,\,\,\,\,\sqrt{49}\cdot\sqrt{3}\)
\(\,\,\,\,\,\,7\sqrt{3}\)
The answer is \(7\sqrt{3}\)
\(\,\,\,\,\,\,\sqrt{147}\)
\(\,\,\,\,\,\,\sqrt{49\cdot3}\)
\(\,\,\,\,\,\,\sqrt{49}\cdot\sqrt{3}\)
\(\,\,\,\,\,\,7\sqrt{3}\)
The answer is \(7\sqrt{3}\)
\(\textbf{15)}\) \(\sqrt{144}\) The answer is \(12\)
\(\,\,\,\,\,\,\sqrt{144}\)
\(\,\,\,\,\,\,\sqrt{12^2}\)
\(\,\,\,\,\,\,12\)
The answer is \(12\)
\(\,\,\,\,\,\,\sqrt{144}\)
\(\,\,\,\,\,\,\sqrt{12^2}\)
\(\,\,\,\,\,\,12\)
The answer is \(12\)
\(\textbf{16)}\) \(\sqrt{192}\) The answer is \(8\sqrt{3}\)
\(\,\,\,\,\,\,\sqrt{192}\)
\(\,\,\,\,\,\,\sqrt{64\cdot3}\)
\(\,\,\,\,\,\,\sqrt{64}\cdot\sqrt{3}\)
\(\,\,\,\,\,\,8\sqrt{3}\)
The answer is \(8\sqrt{3}\)
\(\,\,\,\,\,\,\sqrt{192}\)
\(\,\,\,\,\,\,\sqrt{64\cdot3}\)
\(\,\,\,\,\,\,\sqrt{64}\cdot\sqrt{3}\)
\(\,\,\,\,\,\,8\sqrt{3}\)
The answer is \(8\sqrt{3}\)
\(\textbf{17)}\) \(\sqrt{250}\) The answer is \(5\sqrt{10}\)
\(\,\,\,\,\,\,\sqrt{250}\)
\(\,\,\,\,\,\,\sqrt{25\cdot10}\)
\(\,\,\,\,\,\,\sqrt{25}\cdot\sqrt{10}\)
\(\,\,\,\,\,\,5\sqrt{10}\)
The answer is \(5\sqrt{10}\)
\(\,\,\,\,\,\,\sqrt{250}\)
\(\,\,\,\,\,\,\sqrt{25\cdot10}\)
\(\,\,\,\,\,\,\sqrt{25}\cdot\sqrt{10}\)
\(\,\,\,\,\,\,5\sqrt{10}\)
The answer is \(5\sqrt{10}\)
\(\textbf{18)}\) \(\sqrt{432}\) The answer is \(12\sqrt{3}\)
\(\,\,\,\,\,\,\sqrt{432}\)
\(\,\,\,\,\,\,\sqrt{144\cdot3}\)
\(\,\,\,\,\,\,\sqrt{144}\cdot\sqrt{3}\)
\(\,\,\,\,\,\,12\sqrt{3}\)
The answer is \(12\sqrt{3}\)
\(\,\,\,\,\,\,\sqrt{432}\)
\(\,\,\,\,\,\,\sqrt{144\cdot3}\)
\(\,\,\,\,\,\,\sqrt{144}\cdot\sqrt{3}\)
\(\,\,\,\,\,\,12\sqrt{3}\)
The answer is \(12\sqrt{3}\)
\(\textbf{19)}\) \(\sqrt{500}\) The answer is \(10\sqrt{5}\)
\(\,\,\,\,\,\,\sqrt{500}\)
\(\,\,\,\,\,\,\sqrt{100\cdot5}\)
\(\,\,\,\,\,\,\sqrt{100}\cdot\sqrt{5}\)
\(\,\,\,\,\,\,10\sqrt{5}\)
The answer is \(10\sqrt{5}\)
\(\,\,\,\,\,\,\sqrt{500}\)
\(\,\,\,\,\,\,\sqrt{100\cdot5}\)
\(\,\,\,\,\,\,\sqrt{100}\cdot\sqrt{5}\)
\(\,\,\,\,\,\,10\sqrt{5}\)
The answer is \(10\sqrt{5}\)
\(\textbf{20)}\) \(\sqrt{588}\) The answer is \(14\sqrt{3}\)
\(\,\,\,\,\,\,\sqrt{588}\)
\(\,\,\,\,\,\,\sqrt{196\cdot3}\)
\(\,\,\,\,\,\,\sqrt{196}\cdot\sqrt{3}\)
\(\,\,\,\,\,\,14\sqrt{3}\)
The answer is \(14\sqrt{3}\)
\(\,\,\,\,\,\,\sqrt{588}\)
\(\,\,\,\,\,\,\sqrt{196\cdot3}\)
\(\,\,\,\,\,\,\sqrt{196}\cdot\sqrt{3}\)
\(\,\,\,\,\,\,14\sqrt{3}\)
The answer is \(14\sqrt{3}\)
Challenge Problems
\(\textbf{21)}\) \(2\sqrt{12}+5\sqrt{27}\) The answer is \(19\sqrt{3}\)
\(\,\,\,\,\,\,2\sqrt{12}+5\sqrt{27}\)
\(\,\,\,\,\,\,2\sqrt{4\cdot3}+5\sqrt{9\cdot3}\)
\(\,\,\,\,\,\,2(2\sqrt{3})+5(3\sqrt{3})\)
\(\,\,\,\,\,\,4\sqrt{3}+15\sqrt{3}\)
\(\,\,\,\,\,\,19\sqrt{3}\)
The answer is \(19\sqrt{3}\)
\(\,\,\,\,\,\,2\sqrt{12}+5\sqrt{27}\)
\(\,\,\,\,\,\,2\sqrt{4\cdot3}+5\sqrt{9\cdot3}\)
\(\,\,\,\,\,\,2(2\sqrt{3})+5(3\sqrt{3})\)
\(\,\,\,\,\,\,4\sqrt{3}+15\sqrt{3}\)
\(\,\,\,\,\,\,19\sqrt{3}\)
The answer is \(19\sqrt{3}\)
See Related Pages\(\)
\(\bullet\text{ Multiplying Square Roots}\)
\(\,\,\,\,\,\,\,\, \sqrt{5} \cdot \sqrt{125} …\)
\(\bullet\text{ Dividing Square Roots}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\sqrt{12}}{\sqrt{3}}…\)
\(\bullet\text{ Adding and Subtracting Square Roots}\)
\(\,\,\,\,\,\,\,\,3\sqrt{2}-7\sqrt{2}…\)