Division of Algebraic Expressions

In algebra, we often work with expressions made up of variables and numbers. A monomial is a single term that may include a constant, one or more variables, and their powers — like 7x, −3a²b, or 5xyz².

Just like numbers can be divided, monomials can also be divided — as long as we follow the laws of exponents. When dividing one monomial by another, we:

• Divide the numerical (constant) parts.

• Subtract the exponents of like variables.

This process helps simplify expressions and is a key skill in algebra, especially when solving equations, simplifying fractions, or working with polynomials.

Division of a Monomial by a Monomial

• Rule:To divide one monomial by another:
• Divide monomials using basic arithmetic and the quotient rule for exponents.

Division of a Polynomial by a Monomial

• Rule: Divide each term separately by the same monomial.

(i) `(a^m)/(a^n)` = a^m-n, if m > n and 

(ii) `(a^m)/(a^n)` = `(1) / (a^(n − m))` , if n > m

(i) Division of 12m⁵ by 4m³ = 12m⁵ ÷ 4m³
                                               = `( 3 xx \cancel(4) xx \cancel(m) xx \cancel(m) xx \cancel(m) xx m xx m)/(\cancel(4) xx \cancel(m) xx \cancel(m) xx \cancel(m))`
                                               = 3 × m × m = 3m² 

(ii) `(x^5y^3)/(x^2y^8)`

= `(x^(5-2))/(y^(8-3))`

= `(x^3)/(y^5)` 

(i) Division of `(15x^2y^3)` − `(21x^3y^4)` + `(18x^4y^2)` by  `(3x^2y^2)`

= `(15x^2y^3)/(3x^2y^2)` − `(21x^3y^4)/(3x^2y^2)` +  `(18x^4y^2)/(3x^2y^2)` 

= 5y − 7xy² + 6x² 

• Step 1: Divide the numbers.

• Step 2: Cancel matching letters.

• Step 3: Write what remains.