Lesson 1 · from §60, 61 of the 1910 text
Reducing Fractions to Lowest Terms
Algebraic fractions
An expression like \dfrac{a}{b}, where a and b stand for numbers or polynomials, is an algebraic fraction. Read it "a divided by b" or "a over b." The top is the numerator, the bottom is the denominator, and together they're called the terms of the fraction. Since dividing by zero means nothing, a denominator can never be zero.
Almost everything in this chapter rests on one idea:
Principle. You may multiply or divide the numerator and the denominator by the same expression without changing the value of the fraction.
\frac{8}{4} = \frac{8 \cdot 4}{4 \cdot 4} = \frac{32}{16}, \qquad \frac{18}{30} = \frac{18 \div 6}{30 \div 6} = \frac{3}{5}, \qquad \frac{a}{b} = \frac{an}{bn}.
Each of these is really multiplying or dividing by \frac{n}{n} = 1, which changes the form but never the value.
Lowest terms
A fraction is in lowest terms when nothing but 1 divides both the numerator and the denominator. Cancellation means dividing both terms by a factor they share.
Example. \dfrac{72a^3x^2y^5}{108a^2x^4y^2}. Factor: 72 = 2^3\cdot3^2 and 108 = 2^2\cdot3^3. Cancel 2^2\cdot3^2\,a^2x^2y^2 from both, leaving \dfrac{2ay^3}{3x^2}.
Example. \dfrac{2c^5 - 32c}{4c^6 + 16c^4 - 128c^2}. Factor completely first: \frac{2c(c^2 + 4)(c + 2)(c - 2)}{4c^2(c^2 + 8)(c + 2)(c - 2)}. Cancel 2c(c + 2)(c - 2), and you're left with \dfrac{c^2 + 4}{2c(c^2 + 8)} = \dfrac{c^2 + 4}{2c^3 + 16c}.
Rule. Separate the numerator and the denominator into their prime factors and cancel the factors common to both.
The big warning
Only factors cancel, never terms (the parts joined by + or -). In \dfrac{5 + 2}{6 + 2} = \dfrac{7}{8}, crossing out the 2's would give \dfrac{5}{6}, which is wrong. Likewise nothing cancels in \dfrac{x + a + 4c^2}{y + a + 8c^2}.
Multiplying or dividing both terms is safe. Adding the same number to both terms, or squaring both terms, changes the value. That's a big difference from equations, where you may do the same thing to both sides.
Quick check
\displaystyle \frac{x^2-9}{x^2+5x+6}
From the history of algebra
Our word fraction comes from the Latin fractus, "broken." The horizontal fraction bar was used by Arabic-writing mathematicians in North Africa in the 1100s (al-Hassar is the earliest known), and Leonardo of Pisa (Fibonacci) brought it to Europe in his Liber Abaci of 1202.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Reduce each fraction to lowest terms. Type answers like (x + 1)/(x - 2).
\displaystyle \frac{a^3b}{a^2b^4}
\displaystyle \frac{12x^2y^6}{18xy^7}
\displaystyle \frac{32a^4b^4c}{48a^3b^6c^2}
\displaystyle \frac{45cd^5e^2}{20c^5d^3}
\displaystyle \frac{36a^2b^2}{54a^5bc^3}
\displaystyle \frac{2a}{2a+2}
\displaystyle \frac{3xy+3y^2}{3y^4}
\displaystyle \frac{21d^2+14c}{14c}
\displaystyle \frac{a^2-1}{a^2-2a+1}
\displaystyle \frac{4x^2+4x+1}{4x^2-1}
\displaystyle \frac{2a^2-2b^2}{4a^2-8ab+4b^2}
\displaystyle \frac{18ax^3+9ay}{12bx^3+6by}
\displaystyle \frac{x^2-25}{x^2-x-30}
\displaystyle \frac{c^2-5cd+4d^2}{c^2-16d^2}
\displaystyle \frac{2x^2-2x-180}{2x^2-162}
\displaystyle \frac{21+10x+x^2}{x^2-9}
\displaystyle \frac{4c^3-5c^2-4c+5}{8c^4-10c^3+12c-15}
\displaystyle \frac{x^2-y^2}{(x-y)^2}
\displaystyle \frac{a^2-b^2}{(a-b)^3}
\displaystyle \frac{c^2-d^2}{c^3-d^3}
\displaystyle \frac{y^3-z^3}{(y-z)^3}
\displaystyle \frac{x^3-8}{x^2-4}
\displaystyle \frac{x^6-1}{x^2-1}
\displaystyle \frac{x^4-y^4}{x^6-y^6}
\displaystyle \frac{64x^3+1}{1+8x+16x^2}