Can a monomial have a negative exponent?

Question

I’m struggling with how to convert (x²y) / (2yx^-2) to a standard form monomial since I always find myself left with a negative exponent. However, I thought exponents cannot be negative in a monomial? Is this a rule that differs depending on the definition of a monomial in different places?? Please explain.

Answer 1

A variable with a negative exponent in a denominator can be made positive by bringing it up to the numerator. You can rewrite this as (x²x²y) / (2y). Then you will finish by multiplying x² ⋅ x² = x²⁺². Next a y on top and a y on the bottom cancel because they equal 1 so you are left with just the fraction x⁴/ 2 as your answer. To make any variable with a negative exponent positive just change its position in the fraction from top to bottom OR bottom to top. Look at the examples below to help you understand:

EXAMPLE 1: 3 / (x^-2) = 3x² bring the x^-2 up to make it positive

EXAMPLE 2: (3x^-2) / 5 = 3/5x² bring the x^-2 down to make it positive

Answer 2

Monomial: One or more variables as factors with a non-zero coefficient and non-negative exponents.

(x²y) / (2yx^-2)

x² / (2x^-2), y's reduce

x^{2 - (-2)} / 2, subtract exponent of denominator from exponent of numerator

x⁴ / 2

(1 / 2) x⁴

Answer 3

2x^2y/2yx^-2 simplifies to (2/2)(x^2/x^-2)(y/y)=(1)(x^{2--2})(1)=x^4, with no negative exponent. You are quite right, monomials should not have a negative exponent; this one does not, because the subtraction makes the negative exponent on the bottom into a positive exponent.