Dividing Monomials

While dividing monomials, first divide the numerical coefficients and then handle like variables by subtracting their exponents. This uses the exponent rule, which is the reverse of the exponent rule used in multiplication.

\(15mn5n=(155)(m1)(nn)\)
=(3)(m)(1)
=3m
 

Dividing monomials: Separating numbers and variables. Divide the coefficients, then divide like bases by subtracting exponents. Combine results.

For example: \(14x^2y7x\)
 

Study the coefficients and variables separately.
Divide numbers, write every constant and variable in the expression in the expanded form, grouping common bases: 147=2
We divide the common factor from the numerator and the denominator. For example, \(x^2 ÷ x = x^2-1=x\)
Combine: 2xy
 

When dividing monomials with exponents, we use the exponent rule. Monomials with the same base are divided by subtracting their exponents. Dividing monomials is different from multiplication. In multiplication, we add the exponents of like bases, but in division, we subtract them.   

For example \( y^4y^2\)
Since both terms have the same base y, subtract the exponent.

\(y^4y^2=y^4-2=y^2\)
 

Dividing monomials with negative exponents follows the same exponent rule: subtract the exponents of like bases. However, when subtracting, be careful that negative exponents can result in negative or even positive numbers. In some cases, we rewrite the final answer using positive exponents by applying the rule: 

a-n = 1an 

For example:

\(18x^5 y^26x^2 y^3\)

Divide the coefficients: 186=3

Subtract exponents for like bases.

\(x^5-2=x^3\)

\(y^2-3=y-1\) As the exponent is negative

Combine and simplify negative exponents:

\(3x^3 y-1=3x^3y\)

\(18x^5 y^26x^2 y3=3x^3y\)

While dividing monomials, divide the coefficients and apply the quotient rule of exponents \(\frac{x^m}{x^n} = x^{m - n} \) for the variables. If both monomials have negative coefficients, the answer will have positive coefficients only.
 

For example:
 

\(\frac{-14x^2}{7x} = -14x^{2-1} = -2x \)

(Negative positive = negative)
 

For example:
 

\(\frac{-14x}{-7x} = \frac{-14}{-7} \times \frac{x}{x} = 2 \)
 

(Negative negative = positive)
 

Dividing monomials becomes easy when you follow a few simple rules. These tricks help you simplify expressions quickly and accurately.

• Subtract the exponents of like bases when dividing monomials.
  
   
• Divide the numerical coefficients separately from the variables.
  
   
• Apply the division rule to each variable individually.
  
   
• Rewrite negative exponents as positive by moving them to the denominator.
   

• Cancel common factors to simplify the final expression.

Dividing monomials is used to calculate the force, interest calculations, distance and time, and many more. In this section, we will learn how it is used in these fields. 
 

• Calculating Force: Physicists and engineers use monomials in formulas involving force to design structures, machines, and systems that must withstand specific loads and stresses. Monomials are used in formulae involving force. For example, to find the displacement x in a spring force formula F=kx, divide the total force F by the spring constant (k) to find the displacement (x), x = Fk.
  
   
• Simple Interest Calculation: Use by bankers and finance professionals to calculate interest easily. For example, in the formula I = Prt, dividing or rearranging terms  r=IPt, such as using monomial division to find the unknown efficiently.
  
   
• Time-Speed Formulas: Use by physicists and engineers. This helps in isolating variables for solving rate or time problems. For example, simplify expressions like d=vt or t=d/v in monomial form.
  
   
• Dosage-by-Weight Calculations: Use by pharmacists and healthcare workers to determine precise dosage. For example, if the common dosage is 10 mg per kg, then a patient with 70 kg would receive: dose =10mg/kg × 70kr=700 mg per kg.
  
   
• Profit and Cost Modelling: Used by economists and business analysts to simplify formula manipulation to optimize production. For example, divide monomial cost or revenue terms to study marginal costs.