RD Sharma solution class 8 chapter 6 Algebraic Expressions and Identity Exercise 6.2

Exercise 6.2

Page-6.5

Question 1:

Add the following algebraic expressions:
(i) 3a²b, − 4a²b, 9a²b
(ii) $\frac{2}{3}a, \frac{3}{5}a, -\frac{6}{5}a$
(iii) 4xy² − 7x²y, 12x²y − 6xy², − 3x²y +5xy²
(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) $\frac{11}{2}xy+\frac{12}{5}y+\frac{13}{7}x, -\frac{11}{2}y-\frac{12}{5}x-\frac{13}{7}xy$
(vi) $\frac{7}{2}{x}^3-\frac{1}{2}{x}^2+\frac{5}{3}, \frac{3}{2}{x}^3+\frac{7}{4}{x}^2-x+\frac{1}{3}, \frac{3}{2}{x}^2-\frac{5}{2}x-2$

Answer 1:

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

$$\begin{gathered} 3{\mathrm{a}}^2\mathrm{b}+\left(-4{\mathrm{a}}^2\mathrm{b}\right)+9{\mathrm{a}}^2\mathrm{b} \\ =3{\mathrm{a}}^2\mathrm{b}-4{\mathrm{a}}^2\mathrm{b}+9{\mathrm{a}}^2\mathrm{b} \\ =\left(3-4+9\right){\mathrm{a}}^2\mathrm{b} (\mathrm{Distributive} \mathrm{law}) \\ =8{\mathrm{a}}^2\mathrm{b} \end{gathered}$$

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

$$\begin{gathered} \frac{2}{3}\mathrm{a}+\frac{3}{5}\mathrm{a}+\left(-\frac{6}{5}\mathrm{a}\right) \\ =\frac{2}{3}\mathrm{a}+\frac{3}{5}\mathrm{a}-\frac{6}{5}\mathrm{a} \\ =\left(\frac{2}{3}+\frac{3}{5}-\frac{6}{5}\right)\mathrm{a} ( \mathrm{Distributive} \mathrm{law}) \\ =\frac{1}{15}\mathrm{a} \end{gathered}$$

(iii) To add, we proceed as follows:

$$\begin{gathered} \left(4{\mathrm{xy}}^2-7{\mathrm{x}}^2\mathrm{y}\right)+\left(12{\mathrm{x}}^2\mathrm{y}\right)+\left(-6{\mathrm{xy}}^2\right)+\left(-3{\mathrm{x}}^2\mathrm{y}+5{\mathrm{xy}}^2\right) \\ =4{\mathrm{xy}}^2-7{\mathrm{x}}^2\mathrm{y}+12{\mathrm{x}}^2\mathrm{y}-6{\mathrm{xy}}^2-3{\mathrm{x}}^2\mathrm{y}+5{\mathrm{xy}}^2 \\ =4{\mathrm{xy}}^2-6{\mathrm{xy}}^2+5{\mathrm{xy}}^2-7{\mathrm{x}}^2\mathrm{y}+12{\mathrm{x}}^2\mathrm{y}-3{\mathrm{x}}^2\mathrm{y} ( \mathrm{Collecting} \mathrm{like} \mathrm{terms} ) \\ =3{\mathrm{xy}}^2+2{\mathrm{x}}^2\mathrm{y} (\mathrm{Combining} \mathrm{like} \mathrm{terms}) \end{gathered}$$

(iv) To add, we proceed as follows:

$$\begin{gathered} \left(\frac{3}{2}\mathrm{a}-\frac{5}{4}\mathrm{b}+\frac{2}{5}\mathrm{c}\right)+\left(\frac{2}{3}\mathrm{a}-\frac{7}{2}\mathrm{b}+\frac{7}{2}\mathrm{c}\right)+\left(\frac{5}{3}\mathrm{a}+\frac{5}{2}\mathrm{b}-\frac{5}{4}\mathrm{c}\right) \\ =\frac{3}{2}\mathrm{a}-\frac{5}{4}\mathrm{b}+\frac{2}{5}\mathrm{c}+\frac{2}{3}\mathrm{a}-\frac{7}{2}\mathrm{b}+\frac{7}{2}\mathrm{c}+\frac{5}{3}\mathrm{a}+\frac{5}{2}\mathrm{b}-\frac{5}{4}\mathrm{c} \\ =\frac{3}{2}\mathrm{a}+\frac{2}{3}\mathrm{a}+\frac{5}{3}\mathrm{a}-\frac{5}{4}\mathrm{b}-\frac{7}{2}\mathrm{b}+\frac{5}{2}\mathrm{b}+\frac{2}{5}\mathrm{c}+\frac{7}{2}\mathrm{c}-\frac{5}{4}\mathrm{c} (\mathrm{Collecting} \mathrm{like} \mathrm{terms}) \\ =\frac{23}{6}\mathrm{a}-\frac{9}{4}\mathrm{b}+\frac{53}{20}\mathrm{c} (\mathrm{Combining} \mathrm{like} \mathrm{terms}) \end{gathered}$$

(v) To add, we proceed as follows:

$$\begin{gathered} \left(\frac{11}{2}xy+\frac{12}{5}y+\frac{13}{7}x\right)+\left(-\frac{11}{2}y-\frac{12}{5}x-\frac{13}{7}xy\right) \\ =\frac{11}{2}xy+\frac{12}{5}y+\frac{13}{7}x-\frac{11}{2}y-\frac{12}{5}x-\frac{13}{7}xy \\ =\frac{11}{2}xy-\frac{13}{7}xy +\frac{12}{5}y-\frac{11}{2}y+\frac{13}{7}x-\frac{12}{5}x ( \text{ Collecting like terms)} \\ =\frac{51}{14}xy-\frac{31}{10}y-\frac{19}{35}x (\text{ Combining like terms)} \end{gathered}$$

(vi) To add, we proceed as follows:

$$\begin{gathered} \left(\frac{7}{2}{x}^3-\frac{1}{2}{x}^2+\frac{5}{3}\right)+\left(\frac{3}{2}{x}^3+\frac{7}{4}{x}^2-x+\frac{1}{3}\right)+\left(\frac{3}{2}{x}^2-\frac{5}{2}x-2\right) \\ =\frac{7}{2}{x}^3-\frac{1}{2}{x}^2+\frac{5}{3}+\frac{3}{2}{x}^3+\frac{7}{4}{x}^2-x+\frac{1}{3}+\frac{3}{2}{x}^2-\frac{5}{2}x-2 \\ =\frac{7}{2}{x}^3+\frac{3}{2}{x}^3-\frac{1}{2}{x}^2+\frac{7}{4}{x}^2+\frac{3}{2}{x}^2-x-\frac{5}{2}x+\frac{5}{3}+\frac{1}{3}-2 ( \mathrm{Collecting} \mathrm{like} \mathrm{terms}) \\ =5{\mathrm{x}}^3+\frac{11}{4}{\mathrm{x}}^2-\frac{7}{2}\mathrm{x} ( \mathrm{Combining} \mathrm{like} \mathrm{terms}) \end{gathered}$$

Question 2:

Subtract:
(i) − 5xy from 12xy
(ii) 2a² from − 7a²
(iii) 2a − b from 3a − 5b
(iv) 2x³ − 4x² + 3x + 5 from 4x³ + x² + x + 6
(v) $\frac{2}{3}{y}^3-\frac{2}{7}{y}^2-5 \mathrm{from} \frac{1}{3}{y}^3+\frac{5}{7}{y}^2+y-2$
(vi) $\frac{3}{2}x-\frac{5}{4}y-\frac{7}{2}z \mathrm{from} \frac{2}{3}x+\frac{3}{2}y-\frac{4}{3}z$
(vii) $x^{2}y-\frac{4}{5}x{y}^2+\frac{4}{3}xy \mathrm{from} \frac{2}{3}{x}^{2}y+\frac{3}{2}x{y}^2-\frac{1}{3}xy$
(viii) $\frac{ab}{7}-\frac{35}{3}bc+\frac{6}{5}ac \mathrm{from} \frac{3}{5}bc-\frac{4}{5}ac$

Answer 2:

$$\begin{gathered} \left(\mathrm{i}\right) 12xy-\left(-5xy\right) \\ =12xy+5xy \\ =17xy \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right) -7a^2-\left(2a^2\right) \\ =-7a^2-2a^2 \\ =-9a^2 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right) \left(3a-5b\right)-\left(2a-b\right) \\ =\left(3a-5b\right)-2a+b \\ =3a-5b-2a+b \\ =3a-2a-5b+b \\ =a-4b \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iv}\right) \left(4x^3+x^2+x+6\right)-\left(2x^3-4x^2+3x+5\right) \\ =4x^3+x^2+x+6-2x^3+4x^2-3x-5 \\ =4x^3-2x^3+x^2+4x^2+x-3x+6-5 (\mathrm{Collecting} \mathrm{like} \mathrm{terms}) \\ =2x^3+5x^2-2x+1 (\mathrm{Combining} \mathrm{like} \mathrm{terms}) \end{gathered}$$


$$\begin{gathered} \left(\mathrm{v}\right) \left(\frac{1}{3}{y}^3+\frac{5}{7}{y}^2+y-2\right)-\left(\frac{2}{3}{y}^3-\frac{2}{7}{y}^2-5\right) \\ =\frac{1}{3}{y}^3+\frac{5}{7}{y}^2+y-2-\frac{2}{3}{y}^3+\frac{2}{7}{y}^2+5 \\ =\frac{1}{3}{y}^3-\frac{2}{3}{y}^3+\frac{5}{7}{y}^2+\frac{2}{7}{y}^2+y-2+5 (\mathrm{Collecting} \mathrm{like} \mathrm{terms}) \\ =-\frac{1}{3}{y}^3+y^2+y+3 \left(\text{Combining like terms}\right) \end{gathered}$$


$$\begin{gathered} \left(\mathrm{vi}\right) \left(\frac{2}{3}x+\frac{3}{2}y-\frac{4}{3}z\right)-\left(\frac{3}{2}x-\frac{5}{4}y-\frac{7}{2}z\right) \\ =\frac{2}{3}x+\frac{3}{2}y-\frac{4}{3}z-\frac{3}{2}x+\frac{5}{4}y+\frac{7}{2}z \\ =\frac{2}{3}x-\frac{3}{2}x+\frac{3}{2}y+\frac{5}{4}y-\frac{4}{3}z+\frac{7}{2}z (\mathrm{Collecting} \mathrm{like} \mathrm{terms} ) \\ =-\frac{5}{6}x+\frac{11}{4}y+\frac{13}{6}z (\mathrm{Combining} \mathrm{like} \mathrm{terms}) \end{gathered}$$


$$\begin{gathered} \left(\mathrm{vii}\right) \left(\frac{2}{3}{x}^{2}y+\frac{3}{2}x{y}^2-\frac{1}{3}xy\right)-\left(x^{2}y-\frac{4}{5}x{y}^2+\frac{4}{3}xy\right) \\ =\frac{2}{3}{x}^{2}y+\frac{3}{2}x{y}^2-\frac{1}{3}xy-x^{2}y+\frac{4}{5}x{y}^2-\frac{4}{3}xy \\ =\frac{2}{3}{x}^{2}y-x^{2}y+\frac{3}{2}x{y}^2+\frac{4}{5}x{y}^2-\frac{1}{3}xy-\frac{4}{3}xy (\text{Collecting like terms)} \\ =-\frac{1}{3}{x}^{2}y+\frac{23}{10}x{y}^2 -\frac{5}{3}xy (\text{Combining like terms}) \end{gathered}$$


$$\begin{gathered} \left(\mathrm{viii}\right) \left(\frac{3}{5}bc-\frac{4}{5}ac\right)-\left(\frac{ab}{7}-\frac{35}{3}bc+\frac{6}{5}ac\right) \\ =\frac{3}{5}bc-\frac{4}{5}ac-\frac{ab}{7}+\frac{35}{3}bc-\frac{6}{5}ac \\ =\frac{3}{5}bc+\frac{35}{3}bc-\frac{4}{5}ac-\frac{6}{5}ac-\frac{ab}{7} (\text{Collecting like terms)} \\ =\frac{184}{15}bc-2ac-\frac{ab}{7} (\text{Combining like terms}) \end{gathered}$$

Question 3:

Take away:
(i) $\frac{6}{5}{x}^2-\frac{4}{5}{x}^3+\frac{5}{6}+\frac{3}{2}x \mathrm{from} \frac{x^3}{3}-\frac{5}{2}{x}^2+\frac{3}{5}x+\frac{1}{4}$
(ii) $\frac{5a^2}{2}+\frac{3a^3}{2}+\frac{a}{3}-\frac{6}{5} \mathrm{from} \frac{1}{3}{a}^3-\frac{3}{4}{a}^2-\frac{5}{2}$
(iii) $\frac{7}{4}{x}^3+\frac{3}{5}{x}^2+\frac{1}{2}x+\frac{9}{2} \mathrm{from} \frac{7}{2}-\frac{x}{3}-\frac{x^2}{5}$
(iv) $\frac{y^3}{3}+\frac{7}{3}{y}^2+\frac{1}{2}y+\frac{1}{2} \mathrm{from} \frac{1}{3}-\frac{5}{3}{y}^2$
(v) $\frac{2}{3}ac-\frac{5}{7}ab+\frac{2}{3}bc \mathrm{from} \frac{3}{2}ab-\frac{7}{4}ac-\frac{5}{6}bc$

Answer 3:

(i) The difference is given by:

$$\begin{gathered} \left(\frac{x^3}{3}-\frac{5}{2}{x}^2+\frac{3}{5}x+\frac{1}{4}\right)-\left(\frac{6}{5}{x}^2-\frac{4}{5}{x}^3+\frac{5}{6}+\frac{3}{2}x\right) \\ =\frac{x^3}{3}-\frac{5}{2}{x}^2+\frac{3}{5}x+\frac{1}{4}-\frac{6}{5}{x}^2+\frac{4}{5}{x}^3-\frac{5}{6}-\frac{3}{2}x \end{gathered}$$
$=\frac{x^3}{3}+\frac{4}{5}{x}^3-\frac{5}{2}{x}^2-\frac{6}{5}{x}^2+\frac{3}{5}x-\frac{3}{2}x+\frac{1}{4}-\frac{5}{6}$             (Collecting like terms)
=$\left(\frac{5+12}{15}\right)x^3+\left(\frac{-25-12}{10}\right)x^2+\left(\frac{6-15}{10}\right)x+\left(\frac{6-20}{24}\right)$
$=\frac{17}{15}{x}^3-\frac{37}{10}{x}^2-\frac{9}{10}x-\frac{7}{12}$                                            (Combining like terms)

(ii) The difference is given by:

$$\begin{gathered} \left(\frac{1}{3}{a}^3-\frac{3a^2}{4}-\frac{5}{2}\right)-\left(\frac{5a^2}{2}+\frac{3a^3}{2}+\frac{a}{3}-\frac{6}{5}\right) \\ =\frac{1}{3}{a}^3-\frac{3a^2}{4}-\frac{5}{2}-\frac{5a^2}{2}-\frac{3a^3}{2}-\frac{a}{3}+\frac{6}{5} \end{gathered}$$
$=\frac{1}{3}{a}^3-\frac{3a^3}{2}-\frac{3a^2}{4}-\frac{5a^2}{2}-\frac{a}{3}-\frac{5}{2}+\frac{6}{5}$                      (Collecting like terms)
=$\left(\frac{2-9}{6}\right)a^3+\left(\frac{-3-10}{4}\right)a^2-\frac{a}{3}+\left(\frac{-25+12}{10}\right)$
$=-\frac{7}{6}{a}^3-\frac{13}{4}{a}^2-\frac{a}{3}-\frac{13}{10}$                                             (Combining like terms)

(iii) The difference is given by:

$$\begin{gathered} \left(\frac{7}{2}-\frac{x}{3}-\frac{x^2}{5}\right)-\left(\frac{7x^3}{4}+\frac{3x^2}{5}+\frac{x}{2}+\frac{9}{2}\right) \\ =\frac{7}{2}-\frac{x}{3}-\frac{x^2}{5}-\frac{7x^3}{4}-\frac{3x^2}{5}-\frac{x}{2}-\frac{9}{2} \end{gathered}$$
$=\frac{7}{2}-\frac{9}{2}-\frac{x}{3}-\frac{x}{2}-\frac{x^2}{5}-\frac{3x^2}{5}-\frac{7x^3}{4}$                     (Collecting like terms)
=$\left(\frac{7-9}{2}\right)+\left(\frac{-2-3}{6}\right)x+\left(\frac{-1-3}{5}\right)x^2-\frac{7x^3}{4}$
$=-1-\frac{5x}{6}-\frac{4x^2}{5}-\frac{7x^3}{4}$                                           (Combining like terms )

(iv) The difference is given by:

$$\begin{gathered} \left(\frac{1}{3}-\frac{5}{3}{y}^2\right)-\left(\frac{y^3}{3}+\frac{7y^2}{3}+\frac{y}{2}+\frac{1}{2}\right) \\ =\frac{1}{3}-\frac{5}{3}{y}^2-\frac{y^3}{3}-\frac{7y^2}{3}-\frac{y}{2}-\frac{1}{2} \end{gathered}$$
$=\frac{1}{3}-\frac{1}{2}-\frac{y}{2}-\frac{5}{3}{y}^2-\frac{7y^2}{3}-\frac{y^3}{3}$             ( Collecting like terms)
=$\left(\frac{2-3}{6}\right)-\frac{y}{2}+\left(\frac{-5-7}{3}\right)y^2-\frac{y^3}{3}$
$=-\frac{1}{6}-\frac{y}{2}-4y^2-\frac{y^3}{3}$                               (Combining like terms. )

(v) The difference is given by:

$$\begin{gathered} \left(\frac{3}{2}ab-\frac{7}{4}ac-\frac{5}{6}bc\right)-\left(\frac{2}{3}ac-\frac{5}{7}ab+\frac{2}{3}bc\right) \\ =\frac{3}{2}ab-\frac{7}{4}ac-\frac{5}{6}bc-\frac{2}{3}ac+\frac{5}{7}ab-\frac{2}{3}bc \end{gathered}$$

$=\frac{3}{2}ab+\frac{5}{7}ab-\frac{7}{4}ac-\frac{2}{3}ac-\frac{5}{6}bc-\frac{2}{3}bc$             (Collecting like terms )
=$\left(\frac{21+10}{14}\right)ab+\left(\frac{-21-8}{12}\right)ac+\left(\frac{-5-4}{6}\right)bc$
$=\frac{31}{14}ab-\frac{29}{12}ac-\frac{3}{2}bc$                                         (Combining like terms )

Page-6.6

Question 4:

Subtract 3x − 4y − 7z from the sum of x − 3y + 2z and − 4x + 9y − 11z.

Answer 4:

Let first add the expressions $x-3y+2z \text{and} -4x+9y-11z$. We get:

$\left(x-3y+2z\right)+\left(-4x+9y-11z\right)$    
$=x-3y+2z-4x+9y-11z$
$=x-4x-3y+9y+2z-11z$                     (Collecting like terms)              
$=-3x+6y-9z$                                       (Combining like terms)
                                           
Now, subtracting the expression $3x-4y-7z$ from the above sum; we get:

$$\begin{gathered} \left(-3x+6y-9z\right)-\left(3x-4y-7z\right) \\ =-3x+6y-9z-3x+4y+7z \end{gathered}$$
$=-3x-3x+6y+4y-9z+7z$          (Collecting like terms)
$=-6x+10y-2z$                              (Combining like terms)

Thus, the answer is $-6x+10y-2z$.

Question 5:

Subtract the sum of 3l − 4m − 7n² and 2l + 3m − 4n² from the sum of 9l + 2m − 3n² and − 3l + m + 4n² .....

Answer 5:

We have to subtract the sum of (3l $-$ 4m $-$ 7n²) and (2l + 3m $-$ 4n²) from the sum of (9l + 2m $-$ 3n²) and  ($-$3l + m + 4n²)​

$\left\{\left(9l+2m-3n^2\right)+\left(-3l+m+4n^2\right)\right\}-\left\{\left(3l-4m-7n^2\right)+\left(2l+3m-4n^2\right)\right\}$
$=\left(9l-3l+2m+m-3n^2+4n^2\right)-\left(3l+2l-4m+3m-7n^2-4n^2\right)$              
$=\left(6l+3m+n^2\right)-\left(5l-m-11n^2\right)$                                          (Combining like terms inside the parentheses)
$=6l+3m+n^2-5l+m+11n^2$        
$=6l-5l+3m+m+n^2+11n^2$                                                        (Collecting like terms)
$=l+4m+12n^2$                                                                              (Combining like terms)

Thus, the required solution is $l+4m+12n^2$.

Question 6:

Subtract the sum of 2x − x² + 5 and − 4x − 3 + 7x² from 5.

Answer 6:

We have to subtract the sum of (2x − x² + 5) and (− 4x − 3 + 7x) from 5.

$$\begin{gathered} 5-\left\{\left(2x-x^2+5\right)+\left(-4x-3+7x^2\right)\right\} \\ =5-\left(2x-4x-x^2+7x^2+5-3\right) \\ =5-2x+4x+x^2-7x^2-5+3 \end{gathered}$$
$=5-5+3-2x+4x+x^2-7x^2$                (Collecting like terms)
$=3+2x-6x^2$                                        (Combining like terms)

Thus, the answer is $3+2x-6x^2$.

Question 7:

Simplify each of the following:
(i) x² − 3x + 5 − $\frac{1}{2}$ (3x² − 5x + 7)
(ii) [5 − 3x + 2y − (2x − y)] − (3x − 7y + 9)
(iii) $\frac{11}{2}{x}^{2}y-\frac{9}{4}x{y}^2+\frac{1}{4}xy-\frac{1}{14}{y}^{2}x+\frac{1}{15}y{x}^2+\frac{1}{2}xy$
(iv) $\left(\frac{1}{3}{y}^2-\frac{4}{7}y+11\right)-\left(\frac{1}{7}y-3+2y^2\right)-\left(\frac{2}{7}y-\frac{2}{3}{y}^2+2\right)$
(v) $-\frac{1}{2}{a}^2{b}^{2}c+\frac{1}{3}a{b}^{2}c-\frac{1}{4}ab{c}^2-\frac{1}{5}c{b}^2{a}^2+\frac{1}{6}c{b}^{2}a-\frac{1}{7}{c}^{2}ab+\frac{1}{8}c{a}^{2}b.$

Answer 7:

$$\begin{gathered} \left(\mathrm{i}\right) x^2-3x+5-\frac{1}{2}\left(3x^2-5x+7\right) \\ =x^2-3x+5-\frac{3x^2}{2}+\frac{5x}{2}-\frac{7}{2} \\ =x^2-\frac{3x^2}{2}-3x+\frac{5x}{2}+5-\frac{7}{2} (\mathrm{Collecting} \mathrm{like} \mathrm{terms}) \\ =\left(\frac{1-3}{2}\right)x^2+\left(\frac{-3+5}{2}\right)x+\left(\frac{10-7}{2}\right) \\ =-\frac{x^2}{2}-\frac{x}{2}+\frac{3}{2} \end{gathered}$$

Thus, the answer is $-\frac{x^2}{2}-\frac{x}{2}+\frac{3}{2}$.

$$\begin{gathered} \left(\mathrm{ii}\right) \left[5-3x+2y-\left(2x-y\right)\right]-\left(3x-7y+9\right) \\ =\left[5-3x+2y-2x+y\right]-\left(3x-7y+9\right) \\ =\left[5-5x+3y\right]-\left(3x-7y+9\right) \\ =5-5x+3y-3x+7y-9 \\ =5-9-5x-3x+3y+7y \\ =-4-8x+10y \end{gathered}$$

$\left(\mathrm{iii}\right) \frac{11}{2}{x}^{2}y-\frac{9}{4}x{y}^2+\frac{1}{4}xy-\frac{1}{14}{y}^{2}x+\frac{1}{15}y{x}^2+\frac{1}{2}xy$
$=\frac{11}{2}{x}^{2}y+\frac{1}{15}y{x}^2-\frac{9}{4}x{y}^2-\frac{1}{14}{y}^{2}x+\frac{1}{4}xy+\frac{1}{2}xy$           (Collecting like terms)
=$\left(\frac{165+2}{30}\right)x^{2}y+\left(\frac{-63-2}{28}\right)x{y}^2+\left(\frac{1+2}{4}\right)xy$
$=\frac{167}{30}{x}^{2}y-\frac{65}{28}{y}^{2}x+\frac{3}{2}xy$                                                (Combining like terms)

$$\begin{gathered} \left(\mathrm{iv}\right) \left(\frac{1}{3}{y}^2-\frac{4}{7}y+11\right)-\left(\frac{1}{7}y-3+2y^2\right)-\left(\frac{2}{7}y-\frac{2}{3}{y}^2+2\right) \\ =\frac{1}{3}{y}^2-\frac{4}{7}y+11-\frac{1}{7}y+3-2y^2-\frac{2}{7}y+\frac{2}{3}{y}^2-2 \end{gathered}$$

$=\frac{1}{3}{y}^2-2y^2+\frac{2}{3}{y}^2-\frac{4}{7}y-\frac{1}{7}y-\frac{2}{7}y+11+3-2$        (Collecting like terms)
=$\left(\frac{1-6+2}{3}\right)y^2+\left(\frac{-4-1-2}{7}\right)y+12$
$=-y^2-7y+12$                                                             (Combining like terms)


$\left(\mathrm{v}\right) -\frac{1}{2}{a}^2{b}^{2}c+\frac{1}{3}a{b}^{2}c-\frac{1}{4}ab{c}^2-\frac{1}{5}c{b}^2{a}^2+\frac{1}{6}c{b}^{2}a-\frac{1}{7}{c}^{2}ab+\frac{1}{8}c{a}^{2}b$

$=-\frac{1}{2}{a}^2{b}^{2}c-\frac{1}{5}c{b}^2{a}^2+\frac{1}{3}a{b}^{2}c+\frac{1}{6}c{b}^{2}a-\frac{1}{4}ab{c}^2-\frac{1}{7}{c}^{2}ab+\frac{1}{8}c{a}^{2}b$           (Collecting like terms)
=$\left(\frac{-5-2}{10}\right)a^2{b}^{2}c +\left(\frac{2+1}{6}\right)c{b}^2{a}^2 +\left(\frac{-7-4}{28}\right)c^{2}ab+\frac{1}{8}c{a}^{2}b$
$=-\frac{7}{10}{a}^2{b}^{2}c+\frac{1}{2}a{b}^{2}c-\frac{11}{28}ab{c}^2+\frac{1}{8}{a}^{2}bc$                                                    (Combining like terms)