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🔟elementary algebra review

6.5 Divide Monomials

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Dividing monomials is a key skill in algebra. It involves identifying like terms, canceling common factors, and applying exponent rules. Understanding these concepts helps simplify complex expressions and solve equations efficiently.

Mastering monomial division opens doors to more advanced algebraic techniques. It's crucial for working with polynomials, rational expressions, and solving real-world problems in fields like physics and engineering. Practice makes perfect in this fundamental algebraic operation.

Dividing Monomials

Basics of monomial division

  • Identify like terms (monomials with same variables raised to same powers) to divide monomials
    • In $\frac{15x^2y^3}{5xy^3}$, terms have same variables ($x$ and $y$) but different exponents for $x$
  • Cancel common factors in numerator and denominator
    • $\frac{15x^2y^3}{5xy^3} = \frac{3x^2\cancel{y^3}}{x\cancel{y^3}} = 3x$
  • Divide coefficients and subtract exponents of like bases
    • $\frac{18a^3b^2c}{6a^2bc^2} = 3a^{3-2}b^{2-1}c^{1-2} = 3ab^1c^{-1} = \frac{3ab}{c}$
  • Simplification of monomials often results in a rational expression

Quotient property for monomial divisions

  • States dividing expressions with the same base can subtract the exponents $\frac{a^m}{a^n} = a^{m-n}$
    • $\frac{x^5}{x^2} = x^{5-2} = x^3$
  • Divide coefficients and subtract exponents of like bases when dividing monomials
    • $\frac{12x^3y^2}{6x^2y} = 2x^{3-2}y^{2-1} = 2xy$

Zero exponents in monomial division

  • Non-zero base raised to power of zero equals 1 $a^0 = 1$ ($a \neq 0$)
    • $5^0 = 1$, $x^0 = 1$ ($x \neq 0$)
  • Result is 1 when dividing monomials if exponents of a variable in numerator and denominator are equal
    • $\frac{6x^3y^2}{2x^3y^2} = 3$ because $x^{3-3} = x^0 = 1$ and $y^{2-2} = y^0 = 1$

Quotient to power property

  • States when a quotient is raised to a power, can raise numerator and denominator separately to that power $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
    • $\left(\frac{x^2}{y^3}\right)^4 = \frac{(x^2)^4}{(y^3)^4} = \frac{x^8}{y^{12}}$
  • Apply Quotient to a Power Property first when dividing monomials with the same base raised to a power, then simplify using other exponent properties
    • $\frac{(3x^2)^3}{(9x^5)^2} = \frac{27x^6}{81x^{10}} = \frac{1}{3}x^{6-10} = \frac{1}{3}x^{-4}$

Multiple exponent properties

  • Apply appropriate exponent properties to each variable separately when dividing monomials with multiple variables and exponents
    • $\frac{12x^3y^2z^4}{6x^2y^5z} = 2x^{3-2}y^{2-5}z^{4-1} = 2xy^{-3}z^3$
  • Use Product to a Power Property, Quotient Property, and Quotient to a Power Property as needed to simplify expression step by step
    • $\frac{(2x^3y^2)^2(3xy^4)^3}{(6x^2y^3)^4} = \frac{4x^6y^4 \cdot 27x^3y^{12}}{1296x^8y^{12}} = \frac{108x^9y^{16}}{1296x^8y^{12}} = \frac{1}{12}x^{9-8}y^{16-12} = \frac{1}{12}xy^4$

Negative exponents and reciprocals

  • Division can result in negative exponents when the exponent in the denominator is larger
    • $\frac{x^2}{x^5} = x^{2-5} = x^{-3}$
  • A negative exponent indicates the reciprocal of the base raised to the positive exponent
    • $x^{-3} = \frac{1}{x^3}$
  • When simplifying, move terms with negative exponents to the opposite part of the fraction
    • $2xy^{-3}z^3 = \frac{2xz^3}{y^3}$