128. To Reduce a Fraction to its Lowest Terms.

128. To Reduce a Fraction to its Lowest Terms.

A fraction is in its lowest terms when the numerator and denominator have no common factor. We have, therefore, the following rule:

Resolve the numerator and denominator into their prime factors, and cancel all the common factors.

Reduce the following fractions to their lowest terms:

1. ${\displaystyle \frac{38a^2{b}^3{c}^4}{57a^{3}b{c}^2}}={\displaystyle \frac{2\times 19a^2{b}^3{c}^4}{3\times 19a^{3}b{c}^2}}={\displaystyle \frac{2b^2{c}^2}{3a}}$.

2. ${\displaystyle \frac{a^3-x^3}{a^2-x^2}}={\displaystyle \frac{(a-x)(a^2+ax+x^2)}{(a-x)(a+x)}}={\displaystyle \frac{a^2+ax+x^2}{a+x}}$.

3. ${\displaystyle \frac{a^2+7a+10}{a^2+5a+6}}={\displaystyle \frac{(a+5)(a+2)}{(a+3)(a+2)}}={\displaystyle \frac{a+5}{a+3}}$.

Exercise 42.

Reduce to lowest terms:

1. ${\displaystyle \frac{2a}{6ab}}$.

2. ${\displaystyle \frac{12m^{2}n}{15m{n}^2}}$.

3. ${\displaystyle \frac{21m^2{p}^2}{28m{p}^4}}$.

4. ${\displaystyle \frac{3x^3{y}^{2}z}{6x{y}^3{z}^2}}$.

5. ${\displaystyle \frac{5a^3{b}^3{c}^3}{15c^5}}$.

6. ${\displaystyle \frac{34x^3{y}^4{z}^5}{51x^2{y}^3{z}^5}}$.

7. ${\displaystyle \frac{46m^{2}n{p}^3}{69mn{p}^4}}$.

8. ${\displaystyle \frac{39a^2{b}^3{c}^4}{52a^{5}b{c}^3}}$.

9. ${\displaystyle \frac{58x{y}^4{z}^6}{87x{y}^2{z}^2}}$.