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Express each of the following as a rational number of the form \(\frac{P}{Q}\):
(i) \(\frac{-8}{3}+\frac{-1}{4}+\frac{-11}{6}+\frac{3}{8}-3\)
(ii) \(\frac{6}{7}+1+\frac{-7}{9}+\frac{19}{21}+\frac{-12}{7}\)
(iii) \(\frac{15}{2}+\frac{9}{8}+\frac{-11}{3}+6+\frac{-7}{6}\)
(iv) \(\frac{-7}{4}+0+\frac{-9}{5}+\frac{19}{10}+\frac{11}{14}\)
(v) \(\frac{-7}{4}+\frac{5}{3}+\frac{-1}{2}+\frac{-5}{6}+2\)
Express each of the following as a rational number of the form \(\frac{P}{Q}\):
(i) \(\frac{-8}{3}+\frac{-1}{4}+\frac{-11}{6}+\frac{3}{8}-3\)
(ii) \(\frac{6}{7}+1+\frac{-7}{9}+\frac{19}{21}+\frac{-12}{7}\)
(iii) \(\frac{15}{2}+\frac{9}{8}+\frac{-11}{3}+6+\frac{-7}{6}\)
(iv) \(\frac{-7}{4}+0+\frac{-9}{5}+\frac{19}{10}+\frac{11}{14}\)
(v) \(\frac{-7}{4}+\frac{5}{3}+\frac{-1}{2}+\frac{-5}{6}+2\)
Express each of the following as a rational number of the form p/q, where p and q are integers and q ≠ 0:
(i) 2-3
(ii) (-4)-2
(iii) 1/(3)-2
(iv) (1/2)-5
(v) \((\frac{2}{3})^{-2}\)
Express each of the following as a rational number of the form p/q, where p and q are integers and q ≠ 0:
(i) 2-3
(ii) (-4)-2
(iii) 1/(3)-2
(iv) (1/2)-5
(v) \((\frac{2}{3})^{-2}\)
As (we know that a-n = 1/an)
(i) 2-3 = 1/23 = 1/2×2×2
= 1/8
As (we know that a-n = 1/an)
(ii) (-4)-2 = 1/-42 = 1/-4×-4
= 1/16
As (we know that 1/a-n = an)
(iii) 1/(3)-2 = 32
= 3×3
= 9
As (we know that a-n = 1/an)
(iv) (1/2)-5 = 25 / 15
= 2×2×2×2×2
= 32
As (we know that a-n = 1/an)
(v) (2/3)-2 = 32 / 22
= 3×3 / 2×2
= 9/4
Express each of the following as a rational number of the form \(\frac{q}{p}\), where p and q are integers and q ≠ 0:
(i) \(2^{-3}\)
(ii) \((-4)^{-2}\)
(iii) \(\frac{1}{3^{-2}}\)
(iv) \((\frac{1}{2})^{-5}\)
(v) \((\frac{2}{3})^{-2}\)
Express each of the following as a rational number of the form \(\frac{q}{p}\), where p and q are integers and q ≠ 0:
(i) \(2^{-3}\)
(ii) \((-4)^{-2}\)
(iii) \(\frac{1}{3^{-2}}\)
(iv) \((\frac{1}{2})^{-5}\)
(v) \((\frac{2}{3})^{-2}\)
(i) \(2^{-3}\) = \(\frac{1}{2^{8}}\) = \(\frac{1}{8}\)[Using \(a^{-n} = \frac{1}{a^{n}}\)]
(ii) \((-4)^{-2}\) = \(\frac{1}{-4^{2}}\) = \(\frac{1}{(-4)\times (-4)}\) =\(\frac{1}{16}\)[Using \(a^{-n} = \frac{1}{a^{n}}\) ]
(iii) \(\frac{1}{3^{-2}}\) = \(\frac{3\times 3}{1}\) = \(\frac{9}{1}\)[Using \(\frac{1}{a^{-n}}\) = \(a^{n}\)]
(iv) \((\frac{1}{2})^{-5}\) = \(\frac{2^{5}}{1^{5}}\) = \(\frac{2^{5}}{1}\) = \(\frac{32}{1}\)[Using \(a^{-n}\) = \(\frac{1}{a^{n}}\)]
(v) \((\frac{2}{3})^{-2}\) = \(\frac{3^{2}}{2^{2}}\) = \(\frac{9}{4}\)[Using \(a^{-n}\) = \(\frac{1}{a^{n}}\);\(a^{2}\) = \(a\times a\)]
(i) \(\frac{-8}{3}+(\frac{-1}{4})+(\frac{-11}{6})+\frac{3}{8}-3\)
= \(\frac{-8}{3}-\frac{1}{4}-\frac{11}{6}+\frac{3}{8}-3\)
= \(\frac{-8\times 8}{3\times 8}-\frac{1\times 6}{4\times 6}-\frac{11\times 4}{6\times 4}+\frac{3\times 3}{8\times 3}-\frac{3\times 24}{1\times 24}\)
= \(\frac{-64-6-44+9-72}{24}\)
= \(\frac{-59\times 3}{8\times 3}\)
= \(\frac{-59}{8}\)
(ii) \(\frac{6}{7}+1+(\frac{-7}{9})+(\frac{19}{21})+\frac{-12}{7}\)
= \(\frac{6\times 9}{7\times 9}+\frac{63}{63}-\frac{7\times 7}{9\times 7}-\frac{19\times 3}{21\times 3}-\frac{12\times 9}{7\times 9}\)
= \(\frac{54}{63}+\frac{63}{63}-\frac{49}{63}+\frac{57}{63}-\frac{108}{63}\)
= \(\frac{54+63-49+57-108}{63}\)
= \(\frac{17}{63}\)
(iii) \(\frac{15}{2}+(\frac{9}{8})+(\frac{-11}{3})+6+\frac{-7}{6}\)
= \(\frac{15}{2}+\frac{9}{8}-\frac{11}{3}+6-\frac{7}{6}\)
= \(\frac{15\times 12}{2\times 12}+\frac{9\times 3}{8\times 3}-\frac{11\times 8}{3\times 8}+\frac{6\times 24}{1\times 24}-\frac{7\times 4}{6\times 4}\)
= \(\frac{180+27-88+144-28}{24}\)
= \(\frac{235}{24}\)
(iv) \(\frac{-7}{4}+0+(\frac{-9}{5})+(\frac{19}{10})+\frac{11}{14}\)
= \(\frac{-7}{4}+0-\frac{9}{5}+\frac{19}{10}+\frac{11}{14}\)
= \(\frac{-7\times 35}{4\times 35}-\frac{9\times 28}{5\times 28}+\frac{19\times 14}{10\times 14}+\frac{11\times 10}{14\times 10}\)
= \(\frac{-245-252+266+110}{140}\)
= \(\frac{-121}{140}\)
(v) \(\frac{-7}{4}+(\frac{5}{3})+(\frac{-1}{2})+\frac{-5}{6}+2\)
= \(\frac{-7}{4}+\frac{5}{3}-\frac{1}{2}-\frac{5}{6}+2\)
= \(\frac{-7\times 6}{4\times 6}+\frac{5\times 8}{3\times 8}-\frac{1\times 12}{2\times 12}-\frac{5\times 4}{6\times 4}+\frac{2\times 24}{1\times 24}\)
= \(\frac{-42+40-12-20+48}{24}\)
= \(\frac{14}{24}\)
= \(\frac{7}{12}\)