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Monomials Multiplication & Division

Jul 3, 2025

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Overview

This lesson explains the rules and provides examples for multiplying and dividing monomials, focusing on the product and quotient rules for exponents. It also covers how to handle numerical coefficients, variables with different bases, and negative exponents. The lesson includes step-by-step explorations, practice problems, and guidance on common mistakes.

Multiplying Monomials

  • To multiply monomials with the same base, add their exponents: (x^m \times x^n = x^{m+n}).
    • Example: (x^3 \times x^4 = x^{3+4} = x^7).
  • If the bases are different, write the variables together without combining exponents.
    • Example: (x^4 \times y^6 = x^4y^6).
  • Multiply numerical coefficients as usual, then apply the exponent rule to each variable separately.
    • Example: (9x^3y^4 \times 4x^2y^3 = (9 \times 4)x^{3+2}y^{4+3} = 36x^5y^7).
  • If a variable does not show an exponent, it is understood to have an exponent of 1.
    • Example: (y^4 \times y = y^4 \times y^1 = y^{4+1} = y^5).
  • The product rule applies only when the bases are the same. If the bases are different, do not add exponents.
  • When multiplying several monomials, add the exponents for each variable with the same base, and multiply all numerical coefficients.
    • Example: (x \times x^2 \times x = x^{1+2+1} = x^4).

Examples: Multiplication

  • (x^8 \times x^2 = x^{10}) (add exponents: 8 + 2)
  • (a^2 \times a^3 = a^5) (add exponents: 2 + 3)
  • (2^2 \times 2^5 = 2^7) (add exponents: 2 + 5)
  • (y^4 \times y = y^5) (since (y) is (y^1), add 4 + 1)
  • (x \times x^2 \times x = x^4) (exponents: 1 + 2 + 1)
  • (y^2 \times y \times y = y^4) (exponents: 2 + 1 + 1)
    • Combined: (x^4y^4)
  • (9x^3y^4 \times 4x^2y^3 = 36x^5y^7) (multiply coefficients: 9 × 4 = 36; add exponents for each variable)
  • (-5x^3y^2 \times 3x^3y^4 = -15x^6y^6) (multiply coefficients: -5 × 3 = -15; add exponents)
  • If a variable appears in only one monomial, include it as is in the product.
    • Example: (a^3b \times a^2c = a^{3+2}bc = a^5bc)
  • When multiplying monomials with more than one variable, add exponents for each variable with the same base and write all variables in the product.
    • Example: ((-6a^3bc^4) \times (-3ab c d) = 18a^4b^2c^5d)
      • Multiply coefficients: -6 × -3 = 18
      • Add exponents for each variable: (a^{3+1} = a^4), (b^{1+1} = b^2), (c^{4+1} = c^5), (d) appears only once

Dividing Monomials

  • To divide monomials with the same base, subtract the exponents: (x^m \div x^n = x^{m-n}).
    • Example: (x^5 \div x^3 = x^{5-3} = x^2)
  • If the exponents subtract to zero, the result is 1 ((x^n \div x^n = x^0 = 1)).
  • Divide numerical coefficients as in regular division.
    • Example: (8x^5 \div 2x^2 = (8 \div 2)x^{5-2} = 4x^3)
  • If the numerator’s exponent is smaller than the denominator’s, the result is a negative exponent; move this variable to the denominator and make the exponent positive.
    • Example: (x^2 \div x^5 = x^{2-5} = x^{-3} = 1/x^3)
  • The quotient rule only applies when the bases are the same. If the bases are different, do not subtract exponents.
  • If a variable appears only in the numerator or denominator, include it as is in the answer.
    • Example: (36x^3y^3z \div -4x^2y^3 = -9xz) (z appears only in the numerator)

Examples: Division

  • (m^5n^7o^3 \div m^4n^3o^3 = mn^4)
    • Subtract exponents: (m^{5-4} = m), (n^{7-3} = n^4), (o^{3-3} = o^0 = 1)
  • (8x^5 \div 2x^2 = 4x^3)
    • Divide coefficients: 8 ÷ 2 = 4; subtract exponents: 5-2=3
  • (-21df^5 \div 7d^2e^7f^3 = -3df^2/e^7)
    • Divide coefficients: -21 ÷ 7 = -3
    • (d^{1-2} = d^{-1}) (move to denominator as (1/d)), (f^{5-3} = f^2), (e^{0-7} = e^{-7}) (move to denominator as (1/e^7))
    • Final answer: (-3f^2/(de^7))
  • (-54x^5y^8 \div 9x^3y^7 = -6x^2y)
    • Divide coefficients: -54 ÷ 9 = -6; (x^{5-3} = x^2); (y^{8-7} = y)
  • (36x^3y^3z \div -4x^2y^3 = -9xz)
    • Divide coefficients: 36 ÷ -4 = -9; (x^{3-2} = x); (y^{3-3} = y^0 = 1); z remains
  • If the answer has negative exponents, rewrite as a fraction:
    • Example: (x^{-3}y^{-1} = 1/(x^3y))
  • Only variables with negative exponents are moved to the denominator; variables with positive exponents stay in the numerator.
  • If the numerator and denominator are the same, the result is 1 ((5/5 = 1), (x^3/x^3 = 1)).

Key Terms & Definitions

  • Monomial: An algebraic expression with only one term.
  • Base: The number or variable being raised to a power.
  • Exponent: The number that shows how many times the base is multiplied by itself.
  • Product Rule: When multiplying like bases, add the exponents.
  • Quotient Rule: When dividing like bases, subtract the exponents.
  • Negative Exponent Rule: A variable with a negative exponent moves to the denominator and the exponent becomes positive.
  • Factors: Numbers or variables multiplied together to form a product.

Action Items / Next Steps

  • Complete the worksheet: find the product or quotient of the given monomials.
  • Practice by pausing the video and solving the exercises before checking the answers.
  • Review the provided answer key to check your work and understand any mistakes.
  • Prepare for the next lesson on multiplication and division of binomials and multinomials.
  • Remember: Math is for everyone, and practicing these rules will help you master more complex algebraic expressions.

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