Find which of the variables \(x\), \(y\), \(z\) and \(u\) represent rational numbers and which represent irrational numbers:
(i) \(x^{2}=5\)
(ii) \(y^{2}=9\)
(iii) \(z^{2}=0.04\)
(iv) \(u^{2}=\dfrac{17}{4}\)
Rational numbers: (ii), (iii)
Irrational numbers: (i), (iv)
Find three rational numbers between:
(i) \(-1\) and \(-2\)
(ii) \(0.1\) and \(0.11\)
(iii) \(\dfrac{5}{7}\) and \(\dfrac{6}{7}\)
(iv) \(\dfrac{1}{4}\) and \(\dfrac{1}{5}\)
(i) \(-1.1\), \(-1.2\), \(-1.3\)
(ii) \(0.101\), \(0.102\), \(0.103\)
(iii) \(\dfrac{51}{70}\), \(\dfrac{52}{70}\), \(\dfrac{53}{70}\)
(iv) \(\dfrac{9}{40}\), \(\dfrac{17}{80}\), \(\dfrac{19}{80}\)
Insert a rational number and an irrational number between the following:
(i) \(2\) and \(3\)
(ii) \(0\) and \(0.1\)
(iii) \(\dfrac{1}{3}\) and \(\dfrac{1}{2}\)
(iv) \(-\dfrac{2}{5}\) and \(\dfrac{1}{2}\)
(v) \(0.15\) and \(0.16\)
(vi) \(\sqrt{2}\) and \(\sqrt{3}\)
(vii) \(2.357\) and \(3.121\)
(viii) \(0.0001\) and \(0.001\)
(ix) \(3.623623\) and \(0.484848\)
(x) \(6.375289\) and \(6.375738\)
(i) \(2.1\), \(2.040040004\ldots\)
(ii) \(0.03\), \(0.007000700007\ldots\)
(iii) \(\dfrac{5}{12}\), \(0.414114111\ldots\)
(iv) \(0\), \(0.151151115\ldots\)
(v) \(0.151\), \(0.151551555\ldots\)
(vi) \(1.5\), \(1.585585558\ldots\)
(vii) \(3\), \(3.101101110\ldots\)
(viii) \(0.00011\), \(0.00011331331333\ldots\)
(ix) \(1\), \(1.909009000\ldots\)
(x) \(6.3753\), \(6.375414114111\ldots\)
Represent the following numbers on the number line:
\(7\), \(7.2\), \(-\dfrac{3}{2}\), \(-\dfrac{12}{5}\)
Locate \(\sqrt{5}\), \(\sqrt{10}\) and \(\sqrt{17}\) on the number line.
Represent geometrically the following numbers on the number line:
(i) \(\sqrt{4.5}\)
(ii) \(\sqrt{5.6}\)
(iii) \(\sqrt{8.1}\)
(iv) \(\sqrt{2.3}\)
Express the following in the form \(\dfrac{p}{q}\), where \(p\) and \(q\) are integers and \(q\neq 0\):
(i) \(0.2\)
(ii) \(0.888\ldots\)
(iii) \(5.\overline{2}\)
(iv) \(0.\overline{001}\)
(v) \(0.2555\ldots\)
(vi) \(0.1\overline{34}\)
(vii) \(0.00323232\ldots\)
(viii) \(0.404040\ldots\)
(i) \(\dfrac{1}{5}\)
(ii) \(\dfrac{8}{9}\)
(iii) \(\dfrac{47}{9}\)
(iv) \(\dfrac{1}{999}\)
(v) \(\dfrac{23}{90}\)
(vi) \(\dfrac{133}{990}\)
(vii) \(\dfrac{8}{2475}\)
(viii) \(\dfrac{40}{99}\)
Show that \(0.142857142857\ldots = \dfrac{1}{7}\).
Simplify the following:
(i) \(\sqrt{45}-3\sqrt{20}+4\sqrt{5}\)
(ii) \(\dfrac{\sqrt{24}}{8}+\dfrac{\sqrt{54}}{9}\)
(iii) \(\sqrt[4]{12}\times\sqrt[7]{6}\)
(iv) \(4\sqrt{28}\div 3\sqrt{7}\div \sqrt[3]{7}\)
(v) \(3\sqrt{3}+2\sqrt{27}+\dfrac{7}{\sqrt{3}}\)
(vi) \((\sqrt{3}-\sqrt{2})^{2}\)
(vii) \(\sqrt[4]{81}-8\sqrt[3]{216}+15\sqrt[5]{32}+\sqrt{225}\)
(viii) \(\dfrac{3}{\sqrt{8}}+\dfrac{1}{\sqrt{2}}\)
(ix) \(\dfrac{2\sqrt{3}}{3}-\dfrac{\sqrt{3}}{6}\)
(i) \(\sqrt{5}\)
(ii) \(\dfrac{7\sqrt{6}}{12}\)
(iii) \(168\sqrt{2}\)
(iv) \(\dfrac{8}{3}\)
(v) \(\dfrac{34\sqrt{3}}{3}\)
(vi) \(5-2\sqrt{6}\)
(vii) \(0\)
(viii) \(\dfrac{5}{4}\sqrt{2}\)
(ix) \(\dfrac{\sqrt{3}}{2}\)
Rationalise the denominator of the following:
(i) \(\dfrac{2}{3\sqrt{3}}\)
(ii) \(\dfrac{\sqrt{40}}{\sqrt{3}}\)
(iii) \(\dfrac{3+\sqrt{2}}{4\sqrt{2}}\)
(iv) \(\dfrac{16}{\sqrt{41}-5}\)
(v) \(\dfrac{2+\sqrt{3}}{2-\sqrt{3}}\)
(vi) \(\dfrac{\sqrt{6}}{\sqrt{2}+\sqrt{3}}\)
(vii) \(\dfrac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}\)
(viii) \(\dfrac{3\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}\)
(ix) \(\dfrac{4\sqrt{3}+5\sqrt{2}}{\sqrt{48}+\sqrt{18}}\)
(i) \(\dfrac{2\sqrt{3}}{9}\)
(ii) \(\dfrac{2\sqrt{30}}{3}\)
(iii) \(\dfrac{2+3\sqrt{2}}{8}\)
(iv) \(\sqrt{41}+5\)
(v) \(7+4\sqrt{3}\)
(vi) \(3\sqrt{2}-2\sqrt{3}\)
(vii) \(5+2\sqrt{6}\)
(viii) \(9+2\sqrt{15}\)
(ix) \(\dfrac{9+4\sqrt{6}}{15}\)
Find the values of \(a\) and \(b\) in each of the following:
(i) \(\dfrac{5+2\sqrt{3}}{7+4\sqrt{3}}=a-6\sqrt{3}\)
(ii) \(\dfrac{3-\sqrt{5}}{3+2\sqrt{5}}=a\sqrt{5}-\dfrac{19}{11}\)
(iii) \(\dfrac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}}=2-b\sqrt{6}\)
(iv) \(\dfrac{7+\sqrt{5}}{7-\sqrt{5}}-\dfrac{7-\sqrt{5}}{7+\sqrt{5}}=a+\dfrac{7}{11}\sqrt{5}\,b\)
(i) \(a=11\)
(ii) \(a=\dfrac{9}{11}\)
(iii) \(b=-\dfrac{5}{6}\)
(iv) \(a=0\), \(b=1\)
If \(a=2+\sqrt{3}\), then find the value of \(a-\dfrac{1}{a}\).
\(2\sqrt{3}\)
Rationalise the denominator in each of the following and hence evaluate by taking \(\sqrt{2}=1.414\), \(\sqrt{3}=1.732\) and \(\sqrt{5}=2.236\), up to three places of decimal:
(i) \(\dfrac{4}{\sqrt{3}}\)
(ii) \(\dfrac{6}{\sqrt{6}}\)
(iii) \(\dfrac{\sqrt{10}-\sqrt{5}}{2}\)
(iv) \(\dfrac{\sqrt{2}}{2+\sqrt{2}}\)
(v) \(\dfrac{1}{\sqrt{3}+\sqrt{2}}\)
(i) \(2.309\)
(ii) \(2.449\)
(iii) \(0.463\)
(iv) \(0.414\)
(v) \(0.318\)
Simplify:
(i) \(\left(1^{3}+2^{3}+3^{3}\right)^{1/2}\)
(ii) \(\left(\dfrac{3}{5}\right)^{4}\left(\dfrac{8}{5}\right)^{-12}\left(\dfrac{32}{5}\right)^{6}\)
(iii) \(\left(\dfrac{1}{27}\right)^{-2/3}\)
(iv) \(\left(\left(625\right)^{-1/2}\right)^{-1/4}\)\(^{2}\)
(v) \(\dfrac{9^{1/3}\times 27^{1/2}}{3^{1/2}\times 3^{2}}\)
(vi) \(64^{-1/3}\left(64^{1/3}-64^{2/3}\right)\)
(vii) \(\dfrac{8^{1/3}\times 16^{1/3}}{32^{-1/3}}\)
(i) \(6\)
(ii) \(\dfrac{2025}{64}\)
(iii) \(9\)
(iv) \(5\)
(v) \(3^{-1/3}\)
(vi) \(-3\)
(vii) \(16\)