Worksheet on Simplifying Algebraic Fractions

Practice the simplification provided in the worksheet on simplifying algebraic fractions. The questions are based on simplifying the expressions by cancelling out the common factor from both the numerator and denominator and then simplify by adding or subtracting or multiplying if required.

1. Simplify by adding and subtracting algebraic fractions:

(i) \(\frac{p}{p  -  2} - \frac{p^{2}}{p^{2}  -  4}\)

(ii) \(\frac{3}{a  -  b} - \frac{1}{a  +  b} + \frac{4b}{a^{2}  -  b^{2}}\)

(iii) \(\frac{3}{m  -  3} + \frac{5}{3m  -  9}\)

(iv) \(\frac{z  -  1}{2} + \frac{z  +  3}{5} + \frac{z  +  7}{10}\)

(v) \(\frac{2z  -  1}{3} + \frac{z  -  5}{6} + \frac{z  -  4}{4}\)

(vi) \(\frac{5z  -  1}{8} - \frac{3z  -  2}{7} + \frac{z  -  5}{4}\)

2. Simplify by multiplying and dividing algebraic fractions:

(i) \(\frac{x^{2}  -  4a^{2}}{ax  +  2a^{2}} \times \frac{2a}{x  -  2a}\)

(ii) \(\frac{z^{2}  -  25}{z^{2}  -  4} \times \frac{z  -  2}{z  +  5}\)

(iii) \(\frac{u^{2}  -  16}{u^{2}  -  49} \times \frac{u  +  7}{u  +  4}\)

(iv) \(\frac{m^{2}n^{2}  +  3mn}{4m^{2}  -  1} \div \frac{mn  +  3}{2m  +  1}\)

(v) \(\frac{p^{2}  -  p  -  12}{5p} \times \frac{p^{3}  -  p}{p^{2}  -  9 }\)

(vi) \(\frac{a  +  b}{2a  -  3} \times \frac{a  -  b}{2a  +  3}\)

3. Simplify by cancelling algebraic fractions:

(i) \(\frac{k  -  m}{m} + \frac{k  +  m}{m} - \frac{k^{2}  -  m^{2}}{2km}\)

(ii) \(\frac{z  +  2}{17z} - \frac{z  -  5}{34z} + \frac{z  +  2}{51z}\)

(iii) \(\frac{2x^{2}  -  y^{2}}{x^{2}} - \frac{y^{2}  -  z^{2}}{y^{2}} - \frac{z^{2}  -  x^{2}}{z^{2}}\)

(iv) \(\frac{(d  +  1)^{3}  -  (d  -  1)^{3}}{3d^{3}  +  d}\)

(v) \(\frac{1}{(1  -  a)^{2}} + \frac{2}{1  -  a^{2}} + \frac{1}{(1  +  a)^{2}}\)

(vi) \(\frac{(m^{3}  -  2m)^{2}  -  (m^{2}  -  2)^{2}}{(m  -  1)(m  +  1)(m^{2}  -  2)}\)

Answers for the worksheet on worksheet on simplifying algebraic fractions to its lowest terms are given below to check the exact answers of the above simplification.

Answers:

1. (i) \(\frac{2p}{(p  +  2) (p  –  2)}\)

(ii) \(\frac{2(a  +  4b)}{(a  +  b) (a  –  b)}\)

(iii) \(\frac{14}{3(m  -  3}\)

(iv) \(\frac{4(z  +  1)}{5}\)

(v) \(\frac{13(z  –  2)}{12}\)

(vi) \(\frac{25z  -  61}{56}\)

2. (i) 2

(ii) \(\frac{z  -  5}{z  +  2}\)

(iii) \(\frac{u  -  4}{u  -  7}\)

(iv) \(\frac{mn}{2m  -  1}\)

(v) \(\frac{(p  –  4) (p^{2}  –  1)}{5(p  –  3)}\)

(vi) \(\frac{a^{2}  –  b^{2}}{4a^{2}  -  9}\)

3. (i) \(\frac{k^{2}  +  3m^{2}}{2km}\)

(ii) \(\frac{5z  +  31}{102z}\)

(iii) \(\frac{x^{4}y^{2}  –  y^{4}z^{2}  +  x^{2}z^{4} }{x^{2}y^{2}z^{2}}\)

(iv) \(\frac{2}{d}\)

(v) \(\frac{4}{(1  -  a^{2})^{2}}\)

(vi) 1

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