Square roots are used to find a value that multiplies by itself to make the original number. When simplifying square roots, it is helpful to look for perfect square factors and pull them outside the radical. These problems include simplifying square roots, multiplying radicals, distributing radicals, combining like radicals, and rationalizing denominators.
Practice Problems
\(\textbf{1)}\) \( \displaystyle\sqrt[]{64} \)
The answer is \( 8 \)
\(\,\,\,\,\,\displaystyle\sqrt[]{64}\)
\(\,\,\,\,\,\displaystyle\sqrt[]{8^2}\)
\(\,\,\,\,\,8\)
\(\,\,\,\,\,\)The answer is \(8\)
\(\textbf{2)}\) \( \sqrt{150} \)
The answer is \( 5\sqrt{6} \)
\(\,\,\,\,\,\sqrt{150}\)
\(\,\,\,\,\,\sqrt{25\cdot 6}\)
\(\,\,\,\,\,\sqrt{25}\sqrt{6}\)
\(\,\,\,\,\,5\sqrt{6}\)
\(\,\,\,\,\,\)The answer is \(5\sqrt{6}\)
\(\textbf{3)}\) \( \sqrt{45x^2} \)
The answer is \( 3|x| \sqrt{5} \)
\(\,\,\,\,\,\sqrt{45x^2}\)
\(\,\,\,\,\,\sqrt{9\cdot 5\cdot x^2}\)
\(\,\,\,\,\,\sqrt{9}\sqrt{x^2}\sqrt{5}\)
\(\,\,\,\,\,3|x|\sqrt{5}\)
\(\,\,\,\,\,\)The answer is \(3|x|\sqrt{5}\)
\(\textbf{4)}\) \( \sqrt{20x^2 z^4 w} \)
The answer is \( 2|x| z^2 \sqrt{5w} \)
\(\,\,\,\,\,\sqrt{20x^2z^4w}\)
\(\,\,\,\,\,\sqrt{4\cdot 5\cdot x^2\cdot z^4\cdot w}\)
\(\,\,\,\,\,\sqrt{4}\sqrt{x^2}\sqrt{z^4}\sqrt{5w}\)
\(\,\,\,\,\,2|x|z^2\sqrt{5w}\)
\(\,\,\,\,\,\)The answer is \(2|x|z^2\sqrt{5w}\)
\(\textbf{5)}\) \( \sqrt{15} \cdot \sqrt{35} \)
The answer is \( 5\sqrt{21} \)
\(\,\,\,\,\,\sqrt{15}\cdot\sqrt{35}\)
\(\,\,\,\,\,\sqrt{15\cdot 35}\)
\(\,\,\,\,\,\sqrt{525}\)
\(\,\,\,\,\,\sqrt{25\cdot 21}\)
\(\,\,\,\,\,5\sqrt{21}\)
\(\,\,\,\,\,\)The answer is \(5\sqrt{21}\)
\(\textbf{6)}\) \( \sqrt{5} (\sqrt{10}+2\sqrt{5}) \)
The answer is \( 5\sqrt{2}+10 \)
\(\,\,\,\,\,\sqrt{5}(\sqrt{10}+2\sqrt{5})\)
\(\,\,\,\,\,\sqrt{5}\sqrt{10}+2\sqrt{5}\sqrt{5}\)
\(\,\,\,\,\,\sqrt{50}+2\sqrt{25}\)
\(\,\,\,\,\,5\sqrt{2}+10\)
\(\,\,\,\,\,\)The answer is \(5\sqrt{2}+10\)
\(\textbf{7)}\) \( 2\sqrt{12}+5\sqrt{27} \)
The answer is \( 19\sqrt{3} \)
\(\,\,\,\,\,2\sqrt{12}+5\sqrt{27}\)
\(\,\,\,\,\,2\sqrt{4\cdot 3}+5\sqrt{9\cdot 3}\)
\(\,\,\,\,\,2(2\sqrt{3})+5(3\sqrt{3})\)
\(\,\,\,\,\,4\sqrt{3}+15\sqrt{3}\)
\(\,\,\,\,\,19\sqrt{3}\)
\(\,\,\,\,\,\)The answer is \(19\sqrt{3}\)
\(\textbf{8)}\) \( \sqrt{15} (2+\sqrt{3}) \)
The answer is \( 2\sqrt{15}+3\sqrt{5} \)
\(\,\,\,\,\,\sqrt{15}(2+\sqrt{3})\)
\(\,\,\,\,\,2\sqrt{15}+\sqrt{15}\sqrt{3}\)
\(\,\,\,\,\,2\sqrt{15}+\sqrt{45}\)
\(\,\,\,\,\,2\sqrt{15}+\sqrt{9\cdot 5}\)
\(\,\,\,\,\,2\sqrt{15}+3\sqrt{5}\)
\(\,\,\,\,\,\)The answer is \(2\sqrt{15}+3\sqrt{5}\)
\(\textbf{9)}\) \( \sqrt{5} \cdot \sqrt{125} \)
The answer is \( 25 \)
\(\,\,\,\,\,\sqrt{5}\cdot\sqrt{125}\)
\(\,\,\,\,\,\sqrt{5\cdot 125}\)
\(\,\,\,\,\,\sqrt{625}\)
\(\,\,\,\,\,\sqrt{25^2}\)
\(\,\,\,\,\,25\)
\(\,\,\,\,\,\)The answer is \(25\)
\(\textbf{10)}\) \( 5\sqrt{3}-\sqrt{27} \)
The answer is \( 2\sqrt{3} \)
\(\,\,\,\,\,5\sqrt{3}-\sqrt{27}\)
\(\,\,\,\,\,5\sqrt{3}-\sqrt{9\cdot 3}\)
\(\,\,\,\,\,5\sqrt{3}-3\sqrt{3}\)
\(\,\,\,\,\,2\sqrt{3}\)
\(\,\,\,\,\,\)The answer is \(2\sqrt{3}\)
\(\textbf{11)}\) \(\sqrt{72}\)
The answer is \(6\sqrt{2}\)
\(\,\,\,\,\,\sqrt{72}\)
\(\,\,\,\,\,\sqrt{36\cdot 2}\)
\(\,\,\,\,\,\sqrt{36}\sqrt{2}\)
\(\,\,\,\,\,6\sqrt{2}\)
\(\,\,\,\,\,\)The answer is \(6\sqrt{2}\)
\(\textbf{12)}\) \(\sqrt{48x^4}\)
The answer is \(4x^2\sqrt{3}\)
\(\,\,\,\,\,\sqrt{48x^4}\)
\(\,\,\,\,\,\sqrt{16\cdot 3\cdot x^4}\)
\(\,\,\,\,\,\sqrt{16}\sqrt{x^4}\sqrt{3}\)
\(\,\,\,\,\,4x^2\sqrt{3}\)
\(\,\,\,\,\,\)The answer is \(4x^2\sqrt{3}\)
\(\textbf{13)}\) \(3\sqrt{8}-2\sqrt{18}\)
The answer is \(0\)
\(\,\,\,\,\,3\sqrt{8}-2\sqrt{18}\)
\(\,\,\,\,\,3\sqrt{4\cdot 2}-2\sqrt{9\cdot 2}\)
\(\,\,\,\,\,3(2\sqrt{2})-2(3\sqrt{2})\)
\(\,\,\,\,\,6\sqrt{2}-6\sqrt{2}\)
\(\,\,\,\,\,0\)
\(\,\,\,\,\,\)The answer is \(0\)
\(\textbf{14)}\) \(\sqrt{2}(3\sqrt{8}-\sqrt{50})\)
The answer is \(2\)
\(\,\,\,\,\,\sqrt{2}(3\sqrt{8}-\sqrt{50})\)
\(\,\,\,\,\,\sqrt{2}(3\sqrt{4\cdot 2}-\sqrt{25\cdot 2})\)
\(\,\,\,\,\,\sqrt{2}(6\sqrt{2}-5\sqrt{2})\)
\(\,\,\,\,\,\sqrt{2}(\sqrt{2})\)
\(\,\,\,\,\,2\)
\(\,\,\,\,\,\)The answer is \(2\)
\(\textbf{15)}\) \(\sqrt{98a^2b^4}\)
The answer is \(7|a|b^2\sqrt{2}\)
\(\,\,\,\,\,\sqrt{98a^2b^4}\)
\(\,\,\,\,\,\sqrt{49\cdot 2\cdot a^2\cdot b^4}\)
\(\,\,\,\,\,\sqrt{49}\sqrt{a^2}\sqrt{b^4}\sqrt{2}\)
\(\,\,\,\,\,7|a|b^2\sqrt{2}\)
\(\,\,\,\,\,\)The answer is \(7|a|b^2\sqrt{2}\)
\(\textbf{16)}\) \((\sqrt{3}+\sqrt{5})^2\)
The answer is \(8+2\sqrt{15}\)
\(\,\,\,\,\,(\sqrt{3}+\sqrt{5})^2\)
\(\,\,\,\,\,(\sqrt{3}+\sqrt{5})(\sqrt{3}+\sqrt{5})\)
\(\,\,\,\,\,3+\sqrt{15}+\sqrt{15}+5\)
\(\,\,\,\,\,8+2\sqrt{15}\)
\(\,\,\,\,\,\)The answer is \(8+2\sqrt{15}\)
\(\textbf{17)}\) \(\displaystyle\frac{5}{\sqrt{3}}\)
The answer is \(\displaystyle\frac{5\sqrt{3}}{3}\)
\(\,\,\,\,\,\displaystyle\frac{5}{\sqrt{3}}\)
\(\,\,\,\,\,\displaystyle\frac{5}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}\)
\(\,\,\,\,\,\displaystyle\frac{5\sqrt{3}}{3}\)
\(\,\,\,\,\,\)The answer is \(\displaystyle\frac{5\sqrt{3}}{3}\)
\(\textbf{18)}\) \(4\sqrt{75}+\sqrt{12}\)
The answer is \(22\sqrt{3}\)
\(\,\,\,\,\,4\sqrt{75}+\sqrt{12}\)
\(\,\,\,\,\,4\sqrt{25\cdot 3}+\sqrt{4\cdot 3}\)
\(\,\,\,\,\,4(5\sqrt{3})+2\sqrt{3}\)
\(\,\,\,\,\,20\sqrt{3}+2\sqrt{3}\)
\(\,\,\,\,\,22\sqrt{3}\)
\(\,\,\,\,\,\)The answer is \(22\sqrt{3}\)
\(\textbf{19)}\) \(\sqrt{16x^6y^2}\)
The answer is \(4|x^3||y|\)
\(\,\,\,\,\,\sqrt{16x^6y^2}\)
\(\,\,\,\,\,\sqrt{16}\sqrt{x^6}\sqrt{y^2}\)
\(\,\,\,\,\,4|x^3||y|\)
\(\,\,\,\,\,\)The answer is \(4|x^3||y|\)
\(\textbf{20)}\) \(\displaystyle\frac{3}{2+\sqrt{5}}\)
The answer is \(3\sqrt{5}-6\)
\(\,\,\,\,\,\displaystyle\frac{3}{2+\sqrt{5}}\)
\(\,\,\,\,\,\displaystyle\frac{3}{2+\sqrt{5}}\cdot\frac{2-\sqrt{5}}{2-\sqrt{5}}\)
\(\,\,\,\,\,\displaystyle\frac{3(2-\sqrt{5})}{4-5}\)
\(\,\,\,\,\,\displaystyle\frac{6-3\sqrt{5}}{-1}\)
\(\,\,\,\,\,3\sqrt{5}-6\)
\(\,\,\,\,\,\)The answer is \(3\sqrt{5}-6\)
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