Add the following algebraic expressions:

(i) To add the like terms, we proceed as follows:

$3 a^{2} b+\left(-4 a^{2} b\right)+9 a^{2} b$

$=3 a^{2} b-4 a^{2} b+9 a^{2} b$

$=(3-4+9) a^{2} b$          (Distributive law)

$=8 a^{2} b$

(ii) To add the like terms, we proceed as follows:

$\frac{2}{3} \mathrm{a}+\frac{3}{5} \mathrm{a}+\left(-\frac{6}{5} \mathrm{a}\right)$

$=\frac{2}{3} \mathrm{a}+\frac{3}{5} \mathrm{a}-\frac{6}{5} \mathrm{a}$

$=\left(\frac{2}{3}+\frac{3}{5}-\frac{6}{5}\right) \mathrm{a}$          (Distributive law)

$=\frac{1}{15} \mathrm{a}$   

(iii) To add, we proceed as follows:

$\left(4 x y^{2}-7 x^{2} y\right)+\left(12 x^{2} y\right)+\left(-6 x y^{2}\right)+\left(-3 x^{2} y+5 x y^{2}\right)$

$=4 x y^{2}-7 x^{2} y+12 x^{2} y-6 x y^{2}-3 x^{2} y+5 x y^{2}$

$=4 x y^{2}-6 x y^{2}+5 x y^{2}-7 x^{2} y+12 x^{2} y-3 x^{2} y \quad$ (Collecting like terms )

$=3 x y^{2}+2 x^{2} y \quad$ (Combining like terms)

(iv) To add, we proceed as follows:

$\left(\frac{11}{2} x y+\frac{12}{5} y+\frac{13}{7} x\right)+\left(-\frac{11}{2} y-\frac{12}{5} x-\frac{13}{7} x y\right)$

$=\frac{11}{2} x y+\frac{12}{5} y+\frac{13}{7} x-\frac{11}{2} y-\frac{12}{5} x-\frac{13}{7} x y$

$=\frac{3}{2} \mathrm{a}+\frac{2}{3} \mathrm{a}+\frac{5}{3} \mathrm{a}-\frac{5}{4} \mathrm{~b}-\frac{7}{2} \mathrm{~b}+\frac{5}{2} \mathrm{~b}+\frac{2}{5} \mathrm{c}+\frac{7}{2} \mathrm{c}-\frac{5}{4} \mathrm{c} \quad$ (Collecting like terms)

$=\frac{23}{6} \mathrm{a}-\frac{9}{4} \mathrm{~b}+\frac{53}{20} \mathrm{c}$    (Combining like terms)

(v) To add, we proceed as follows:

$\left(\frac{11}{2} x y+\frac{12}{5} y+\frac{13}{7} x\right)+\left(-\frac{11}{2} y-\frac{12}{5} x-\frac{13}{7} x y\right)$

$=\frac{11}{2} x y+\frac{12}{5} y+\frac{13}{7} x-\frac{11}{2} y-\frac{12}{5} x-\frac{13}{7} x y$

$=\frac{11}{2} x y-\frac{13}{7} x y+\frac{12}{5} y-\frac{11}{2} y+\frac{13}{7} x-\frac{12}{5} x \quad$ (Collecting like terms)

$=\frac{51}{14} x y-\frac{31}{10} y-\frac{19}{35} x \quad$ (Combining like terms)

(vi) To add, we proceed as follows:

$\left(\frac{7}{2} x^{3}-\frac{1}{2} x^{2}+\frac{5}{3}\right)+\left(\frac{3}{2} x^{3}+\frac{7}{4} x^{2}-x+\frac{1}{3}\right)+\left(\frac{3}{2} x^{2}-\frac{5}{2} x-2\right)$

$=\frac{7}{2} x^{3}-\frac{1}{2} x^{2}+\frac{5}{3}+\frac{3}{2} x^{3}+\frac{7}{4} x^{2}-x+\frac{1}{3}+\frac{3}{2} x^{2}-\frac{5}{2} x-2$

$=\frac{7}{2} x^{3}+\frac{3}{2} x^{3}-\frac{1}{2} x^{2}+\frac{7}{4} x^{2}+\frac{3}{2} x^{2}-x-\frac{5}{2} x+\frac{5}{3}+\frac{1}{3}-2 \quad$ ( Collecting like terms)

$=5 \mathrm{x}^{3}+\frac{11}{4} \mathrm{x}^{2}-\frac{7}{2} \mathrm{x} \quad$ (Combining like terms)