GoalBring algebraic fractions to a common denominator.
New words
common denominator · umumiy maxrajadditional factor · qo‘shimcha ko‘paytuvchifactoring the denominator · maxrajni ajratishbasic property of a fraction · kasrning asosiy xossasi
Explanation
Bringing fractions to a common denominator means replacing them by fractions with identical denominators. The procedure: 1) factor the denominators of the given fractions; 2) take as common denominator the simplest product divisible by every denominator (each factor to its highest power); 3) divide the common denominator by each denominator to find the additional factor; 4) multiply the numerator and denominator of each fraction by its additional factor. For example, for 5/(6a²b) and 3/(4ab²) the common denominator is 12a²b² and the additional factors are 2b and 3a. If the denominators are polynomials we factor them first: x² - 4 = (x - 2)(x + 2). If the denominators contain opposite factors (x - y and y - x), we change the sign of one of them.
Worked examples
5/(6a²b) = 10b/(12a²b²); 3/(4ab²) = 9a/(12a²b²). The additional factors: 12a²b² : 6a²b = 2b and 12a²b² : 4ab² = 3a.
1/(x² - 4) and 3/(x + 2): x² - 4 = (x - 2)(x + 2), so the common denominator is (x - 2)(x + 2). Result: 1/((x - 2)(x + 2)) and 3(x - 2)/((x - 2)(x + 2)).
Class activity
“Denominator lifts”: each fraction rides a “lift” that raises its denominator to the common denominator. Pupils find the additional-factor card for every fraction and “run” the lift.
Practice
1
Bring 1/(2a) and 3/(5b) to a common denominator.
5b/(10ab) and 6a/(10ab)
2
Bring 4/(m² - n²) and 2/(m + n) to a common denominator.
4/((m - n)(m + n)) and 2(m - n)/((m - n)(m + n))
3
If the numerical coefficients of the denominators are 12 and 18, what is their least common multiple?
36
4
Why do we factor the denominators before finding the common denominator?
The factors show which ones are shared, so the common denominator comes out in its simplest form.