Practice Problems
Simplify
\(\textbf{1)}\) \( \sqrt{24} \) The answer is \( 2\sqrt{6} \)
\(\,\,\,\,\, \sqrt{24} \)
\(\,\,\,\,\, \sqrt{4} \cdot \sqrt{6} \)
\(\,\,\,\,\, 2\sqrt{6} \)
\(\,\,\,\,\, \sqrt{24} \)
\(\,\,\,\,\, \sqrt{4} \cdot \sqrt{6} \)
\(\,\,\,\,\, 2\sqrt{6} \)
\(\textbf{2)}\) \( \sqrt{48} \) The answer is \( 4\sqrt{3} \)
\(\,\,\,\,\, \sqrt{48} \)
\(\,\,\,\,\, \sqrt{16} \cdot \sqrt{3} \)
\(\,\,\,\,\, 4\sqrt{3} \)
\(\,\,\,\,\, \sqrt{48} \)
\(\,\,\,\,\, \sqrt{16} \cdot \sqrt{3} \)
\(\,\,\,\,\, 4\sqrt{3} \)
\(\textbf{3)}\) \( \sqrt{120} \) The answer is \( 2\sqrt{30} \)
\(\,\,\,\,\, \sqrt{120} \)
\(\,\,\,\,\, \sqrt{4} \cdot \sqrt{30} \)
\(\,\,\,\,\, 2\sqrt{30} \)
\(\,\,\,\,\, \sqrt{120} \)
\(\,\,\,\,\, \sqrt{4} \cdot \sqrt{30} \)
\(\,\,\,\,\, 2\sqrt{30} \)
\(\textbf{4)}\) \( \sqrt{32} \) The answer is \( 4\sqrt{2} \)
\(\,\,\,\,\, \sqrt{32} \)
\(\,\,\,\,\, \sqrt{16} \cdot \sqrt{2} \)
\(\,\,\,\,\, 4\sqrt{2} \)
\(\,\,\,\,\, \sqrt{32} \)
\(\,\,\,\,\, \sqrt{16} \cdot \sqrt{2} \)
\(\,\,\,\,\, 4\sqrt{2} \)
\(\textbf{5)}\) \( \sqrt{180} \) The answer is \( 6\sqrt{5} \)
\(\,\,\,\,\, \sqrt{180} \)
\(\,\,\,\,\, \sqrt{36} \cdot \sqrt{5} \)
\(\,\,\,\,\, 6\sqrt{5} \)
\(\,\,\,\,\, \sqrt{180} \)
\(\,\,\,\,\, \sqrt{36} \cdot \sqrt{5} \)
\(\,\,\,\,\, 6\sqrt{5} \)
\(\textbf{6)}\) \( \sqrt{6}\cdot \sqrt{15} \) The answer is \( 3\sqrt{10} \)
\(\,\,\,\,\, \sqrt{6}\cdot \sqrt{15} \)
\(\,\,\,\,\, \sqrt{6 \cdot 15}\)
\(\,\,\,\,\, \sqrt{2 \cdot 3 \cdot 3 \cdot 5}\)
\(\,\,\,\,\, \sqrt{9} \cdot \sqrt{10}\)
\(\,\,\,\,\, 3\sqrt{10} \)
\(\,\,\,\,\, \sqrt{6}\cdot \sqrt{15} \)
\(\,\,\,\,\, \sqrt{6 \cdot 15}\)
\(\,\,\,\,\, \sqrt{2 \cdot 3 \cdot 3 \cdot 5}\)
\(\,\,\,\,\, \sqrt{9} \cdot \sqrt{10}\)
\(\,\,\,\,\, 3\sqrt{10} \)
\(\textbf{7)}\) \( -2\sqrt{18} \cdot 5\sqrt{8} \) The answer is \( -120 \)
\(\textbf{8)}\) \( \sqrt{48a^6 b^5} \) The answer is \( 4a^3 b^2 \sqrt{3b} \)
\(\textbf{9)}\) \( (6+4\sqrt{2})+(3-2\sqrt{2}) \) The answer is \( 9+2\sqrt{2} \)
\(\textbf{10)}\) \( (5-3\sqrt{3})-(3+2\sqrt{3}) \) The answer is \( 2-5\sqrt{3} \)
\(\textbf{11)}\) \( \displaystyle\frac{2}{3+4\sqrt{2}} \) The answer is \( \displaystyle\frac{-6+8\sqrt{2}}{23} \)
\(\textbf{12)}\) \( \displaystyle\frac{3-2\sqrt{2}}{\sqrt{6}} \) The answer is \( \displaystyle\frac{3\sqrt{6}-4\sqrt{3})}{6} \)
\(\textbf{13)}\) \( \displaystyle\frac{6+\sqrt{5}}{3+2\sqrt{5}} \) The answer is \( \displaystyle\frac{9\sqrt{5}-8}{11}\)
\(\textbf{14)}\) \( (5+\sqrt{6})(5-\sqrt{6}) \) The answer is \( 19 \)
\(\textbf{15)}\) \( (4+3\sqrt{7})(5-6\sqrt{7}) \) The answer is \( -106-9\sqrt{7} \)
\(\textbf{16)}\) Solve for x in \( 5x^2-25=0 \) The answer is \( x=\pm\sqrt{5} \)
\(\textbf{17)}\) Find the conjugate of \( 3+2\sqrt{2} \) The complex conjugate is \( 3-2\sqrt{2} \)
In Summary
Irrational numbers are numbers that cannot be expressed as a ratio of two integers, or as a simple fraction. This means that their decimal representation is non-terminating and non-repeating. Examples of irrational numbers include \(\sqrt{2}\) and \(pi\).
Irrational numbers are typically introduced in a high school or college algebra or pre-calculus class.
Irrational numbers were first discovered by the ancient Greeks, who were fascinated by the concept of numbers that could not be expressed as simple fractions.
One example of irrational numbers occurs in geometry, where they are used to represent the lengths of certain geometric figures, such as the diagonal of a square or the circumference of a circle.