To divide a monomial by another monomial, follow the steps below.

(1)               Divide the signs of the monomials.

(2)               Divide the coefficients of the monomials.

(3)               Divide the like variables. Use the rule for the division of variables with exponents. To do this, subtract the exponents of like variables.

 Remember that when a number or a variable has an exponent of zero, the value of that exponent or variable is 1.

A simple mnemonic to help you deal with negative exponents is to think of fractions as a 2-story apartment house. Someone lives on the top story and someone lives on the bottom story. If they have a negative exponent, they are negative or unhappy. When they move to the other apartment, they become happy, and their exponent becomes positive. The exponent applies only to the number or letter that it is beside.

Example: Here the letter a and the number 3 each have negative exponents. They are unhappy and want to move to the other apartment. Once moved, they will be happy: 

Practice 7. Divide.

(a)   32x⁴y⁵z³ ÷ 8x²y²z²

(b)   –65mn⁶x⁴ ÷ 13mn³x²

(c)   (–121ab⁵c⁴) ÷ (–11ab³c²)

Solution.

(a)   32x⁴y⁵z³ ÷ 8x²y²z² = 4(x^{4 – 2})(y^{5 – 2})(z^{3 – 2})

   = 4x²y³z

(b)   –65mn⁶x⁴ ÷ 13mn³x² = (–65 ÷ 13)(m^{1 – 1})(n^{6 – 3})(x^{4 – 2})

       = –5n³x²

(c)   (–121ab⁵c⁴) ÷ (–11ab³c²) = [(–121) ÷ (–11)](a^{1 – 1})(b^{5 – 3})(c^{4 – 2})

   = 11b²c²

Practical Exercise 2. Divide.

(a) –x⁶y⁵z⁷ ÷ 8x²y²z²

(b) –625m⁴n⁶x⁴ ÷ 25m⁴n³x⁴

(c) (–120a⁴b⁵c⁴) ÷ (–24ab³c²)

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