Dividing Monomials

Just as 2 \div 3 is written \tfrac{2}{3}, a \div b is written as the fraction \tfrac{a}{b} and can't be simplified further. Likewise 4a \div 2 = 2a, but 12 \div 4y = \tfrac{3}{y}, and -12a \div 6b = -\tfrac{2a}{b}.

Subtracting exponents

Since a^5 = a \cdot a \cdot a \cdot a \cdot a and a^2 = a \cdot a, a^5 \div a^2 = \frac{a \cdot a \cdot a \cdot a \cdot a}{a \cdot a} = a^3 = a^{5-2}. Likewise 6b^4y^5 \div 2b^2y^3 = 3b^2y^2.

Principle. The exponent of any letter in the quotient equals its exponent in the dividend minus its exponent in the divisor: \;n^a \div n^b = n^{a-b}.

(What this means when b = a, or when b is greater than a, comes later.) The sign rules are the same as before: like signs give +, unlike signs give -.

A letter with the same exponent in dividend and divisor cancels completely: ax^2 \div x^2 = a.

Rule. Divide the coefficients, using the rule of signs. After that, write each letter of the dividend (except ones that cancel) with its exponent in the dividend minus its exponent in the divisor. Any letters found only in the divisor go underneath as a denominator.

Typing tip: type fractions with a slash, like -4a/b, and letter exponents in parentheses, like x^(2a).