Add the following algebraic expressions
(i) $ 3 a^{2} b,-4 a^{2} b, 9 a^{2} b $
(ii) $ \frac{2}{3} a, \frac{3}{5} a,-\frac{6}{5} a $
(iii) $ 4 x y^{2}-7 x^{2} y, 12 x^{2} y-6 x y^{2},-3 x^{2} y+5 x y^{2} $
(iv) $ \frac{3}{2} a-\frac{5}{4} b+\frac{2}{5} c, \frac{2}{3} a-\frac{7}{2} b+\frac{7}{2} c, \frac{5}{3} a+ $ $ \frac{5}{2} b-\frac{5}{4} c $
(v) $ \frac{11}{2} x y+\frac{12}{5} y+\frac{13}{7} x,-\frac{11}{2} y-\frac{12}{5} x-\frac{13}{7} x y $
(vi) $ \frac{7}{2} x^{3}-\frac{1}{2} x^{2}+\frac{5}{3}, \frac{3}{2} x^{3}+\frac{7}{4} x^{2}-x+\frac{1}{3} $ $ \frac{3}{2} x^{2}-\frac{5}{2} x-2 $
To do:
We have to add the given algebraic expressions.
Solution:
(i) $3 a^{2} b+(-4 a^{2} b)+9 a^{2} b=a^{2} b(3-4+9)$
$=8 a^{2} b$
(ii) $\frac{2}{3} a+\frac{3}{5} a+(\frac{-6}{5} a)=(\frac{2}{3}+\frac{3}{5}-\frac{6}{5}) a$
$=\frac{10+9-18}{15} a$ (LCM of 3 and 5 is 15)
$=\frac{19-18}{15} a$
$=\frac{1}{15} a$
(iii) $4 x y^{2}-7 x^{2} y+12 x^{2} y-6 x y^{2}-3 x^{2} y+5 x y^{2}=4 x y^{2}-6 x y^{2}+5 x y^{2}-7 x^{2} y+12 x^{2} y-3 x^{2} y$
$=(4-6+5) x y^{2}+(-7+12-3) x^{2} y$
$=3 x y^{2}+2 x^{2} y$
(iv) $\frac{3}{2} a-\frac{5}{4} b+\frac{2}{5} c+\frac{2}{3} a-\frac{7}{2} b+\frac{7}{2} c+\frac{5}{3} a+\frac{5}{2} b-\frac{5}{4} c=\frac{3}{2} a+\frac{2}{3} a+\frac{5}{3} a-\frac{5}{4} b-\frac{7}{2} b+\frac{5}{2} b+\frac{2}{5} c+\frac{7}{2} c-\frac{5}{4} c$
$=(\frac{9 a+4 a+10 a}{6})+(\frac{-5 b-14 b+10 b}{4})+(\frac{8 c+70 c-25 c}{20})$
$=\frac{23}{6} a-\frac{9}{4} b+\frac{53}{20} c$
(v) $\frac{11}{2} x y+\frac{12}{5} y+\frac{13}{7} x-\frac{11}{2} y-\frac{12}{5} x-\frac{13}{7} x y=\frac{11}{2} x y-\frac{13}{7} x y+\frac{12}{5} y-\frac{11}{2} y+\frac{13}{7} x-\frac{12}{5} x$
$=(\frac{77 x y-26 x y}{14})+(\frac{24 y-55 y}{10})+(\frac{65 x-84 x}{35})$
$=(\frac{51}{14} x y)+(\frac{-31}{10} y)+(\frac{-19}{35} x)$
$=\frac{51}{14} x y-\frac{31}{10} y-\frac{19}{35} x$
$=\frac{51}{14} x y-\frac{19}{35} x-\frac{31}{10} y$
(vi) $\frac{7}{2} x^{3}-\frac{1}{2} x^{2}+\frac{5}{3}+\frac{3}{2} x^{3}+\frac{7}{4} x^{2}-x+\frac{1}{3}+\frac{3}{2} x^{2}-\frac{5}{2} x-2$
$=\frac{7}{2} x^{3}+\frac{3}{2} x^{3}-\frac{1}{2} x^{2}+\frac{7}{4} x^{2}+\frac{3}{2} x^{2}-\frac{x}{1}-\frac{5}{2} x+\frac{5}{3}+\frac{1}{3}-2$
$=(\frac{7 x^{3}+3 x^{3}}{2})+(\frac{-2 x^{2}+7 x^{2}+6 x^{2}}{4})+(\frac{-2 x-5 x}{2})+(\frac{5+1-6}{3})$
$=\frac{10}{2} x^{3}+\frac{11}{4} x^{2}-\frac{7}{2} x+0$
$=5 x^{3}+\frac{11}{4} x^{2}-\frac{7}{2} x$
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