Opening the book…
Identify which of the following numbers are irrational: \(\sqrt{441}, 4.33, \sqrt{70}, 5+9 \pi\)
Identify which of the following numbers are irrational: \(\frac{13}{\sqrt{46}}, 4+13 \pi, \sqrt{144}, \frac{5}{9}\)
For the following exercises, simplify the square root by expressing it in lowest terms.
\(\sqrt{12}\)
\(\sqrt{75}\)
\(\sqrt{605}\)
\(\sqrt{45}\)
\(\sqrt{112}\)
\(\sqrt{396}\)
\(\sqrt{2,940}\)
\(\sqrt{2,400}\)
\(\sqrt{3,240}\)
\(\sqrt{5,472}\)
For the following exercises, perform the arithmetic operations without a calculator, if possible. If it is not possible, state why.
\(4 \sqrt{3}+2 \sqrt{3}\)
\(8 \sqrt{5}+3 \sqrt{5}\)
\(9 \sqrt{7}-15 \sqrt{7}\)
\(\sqrt{13}-15 \sqrt{13}\)
\(8 \pi-13 \sqrt{2}\)
\(7 \sqrt{5}+6 \sqrt{14}\)
\(7.2 \pi+8.6 \pi\)
\(14.5 \pi-5.8 \pi\)
\(19.8 \sqrt{12}-6.1 \sqrt{3}\)
\(7.3 \sqrt{45}-6.8 \sqrt{20}\)
\((4 \sqrt{15}) \times(3 \sqrt{10})\)
\((7 \sqrt{33}) \times(8 \sqrt{66})\)
\((4.5 \sqrt{154}) \div(3 \sqrt{77})\)
\((70 \sqrt{30}) \div(14 \sqrt{6})\)
For the following exercises, rationalize the denominators of the fractions, and then simplify.
\(\frac{3}{\sqrt{2}}\)
\(\frac{8}{\sqrt{6}}\)
\(\frac{5}{\sqrt{13}}\)
\(\frac{9}{\sqrt{35}}\)
\(\frac{12}{\sqrt{40}}\)
\(\frac{20}{\sqrt{150}}\)
Determine the conjugate of \(5+\sqrt{6}\).
Determine the conjugate of \(10-\sqrt{13}\).
Determine the conjugate of \(4+3 \sqrt{5}\).
Determine the conjugate of \(5 \sqrt{15}+8 \sqrt{13}\).
Find the product of \(3+2 \sqrt{7}\) and its conjugate.
Find the product of \(2 \sqrt{3-} 5\) and its conjugate.
For the following exercises, rationalize the denominator of the fraction, and then simplify the fraction.
\(\frac{4}{1+\sqrt{3}}\)
\(\frac{6}{5+\sqrt{7}}\)
\(\frac{-4}{5-\sqrt{10}}\)
\(\frac{10}{\sqrt{6}+7}\)
3.5.0: Exercises is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.
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