Worksheet on Algebraic Expressions to the Lowest Terms

Practice the worksheet on algebraic expressions to the lowest terms. The questions are based on simplifying by cancelling the algebraic fractions to reduce them to their simplest form.

1. Reduce the algebraic expressions to its simplest form:

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

(ii) \(\frac{1}{2b^{2}  +  b  -  6} + \frac{1}{3b^{2}  +  5b  -  2}\)

(iii) \(\frac{2(a  -  3)}{a^{2}  -  5a  +  6} + \frac{3(a  -  1)}{a^{2}  -  4a  +  3} + \frac{5(a  -  2)}{a^{2}  -  3a  +  2}\)

(iv) \(\frac{u}{9} + \frac{2}{3} + \frac{4}{u  -  6} - \frac{2}{3}\frac{1}{1 - \frac{6}{u}}\)

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

(vi) \(\frac{x^{2}  -  yz}{yz} - \frac{xz  -  y^{2}}{xz} - \frac{xy  -  z^{2}}{xy}\)

2. Reduce by multiplying and dividing the algebraic fractions to its lowest term:

(i) \(\frac{z^{2}  -  121}{z^{2}  -  4} \div \frac{z  +  11}{z  +  2}\)

(ii) \(\frac{x  -  3y}{x  +  2y} \div \frac{x^{2}  -  9y^{2}}{x^{2}  -  4y^{2}}\)

(iii) \(\frac{a^{2}  -  2a}{a^{2}  +  3a  -  10} \div \frac{a^{2}  +  4a  -  21}{a^{2}  +  2a  -  15}\)

(iv) \(\frac{14k^{2}  -  7k}{12k^{3}  +  24k^{2}} \div \frac{2k  -  1}{k^{2}  +  2k}\)

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

(vi) \(\frac{n^{2}  -  15n  +  4}{n^{2}  -  7n  +  10} \times \frac{n^{2}  -  n  -  2}{n^{2}  +  2n  -  3} \div \frac{n^{2}  -  5n  +  4}{n^{2}  +  8n  +  15}\)

3. Simplify by reducing to its simplest form:

(i) \(\frac{2z  -  3}{9} - \frac{z  +  2}{6} + \frac{5z  +  8}{12}\)

(ii) \(\frac{m  -  7}{15} + \frac{m  -  9}{25} - \frac{m  +  3}{45}\)

(iii) \(\frac{2k  +  5}{k} - \frac{k  +  3}{2k} - \frac{27}{8k^{2}}\)

(iv) \(\frac{x  -  y}{xy} + \frac{y  -  z}{yz} + \frac{z  -  x}{zx}\)

(v) \(\frac{m  -  2n}{2m} - \frac{m  -  5n}{4m} + \frac{m  +  7n}{8m}\)

(vi) \(\frac{q  +  r}{2p}  +  \frac{r  +  p}{4q}  -  \frac{p  -  q}{3r}\)

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

Answers:

1. (i) \(\frac{1}{(z  –  1) (z  –  2) (z  –  3)}\)

(ii) \(\frac{5b  –  4}{(2b  –  3) (b  +  2) (3b  –  1)}\)

(iii) \(\frac{2(5a^{2}  –  21a  +  21)}{(a  –  1) (a  –  2) (a  –  3)}\)

(iv) \(\frac{u}{9}\)

(v) \(\frac{2a  -  b}{a^{2}  -  b^{2}}\)

(vi) \(\frac{x^{3}  +  y^{3}  +  z^{3}  –  3xyz }{xyz}\)

2. (i) \(\frac{z  -  11}{z  -  2}\)

(ii) \(\frac{x  –  2y}{x  +  3y}\)

(iii) \(\frac{a}{a  +  7}\)

(iv) \(\frac{7}{12}\)

(v) \(\frac{mn}{2m  -  1}\)

(vi) \(\frac{(n^{2}  –  15n  +  4) (n  +  1) (n  +  5)}{(n  -  5) (n  -  4) (n  -  1) (n  -  1)}\)

3. (i) \(\frac{17z}{36}\)

(ii) \(\frac{19m  -  201}{225}\)

(iii) \(\frac{12k^{2}  +  28k  -  27}{8k^{2}}\)

(iv) 0

(v) \(\frac{3(m  +  3n)}{8m}\)

(vi) \(\frac{6q^{2}r  +  6qr^{2}  +  3pr^{2}  +  3p^{2}r  –  4p^{2}q  +  4pq^{2}}{12pqr}\)

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