6.5 Divide Monomials

Dividing Monomials

Dividing monomials comes down to two core moves: divide the coefficients (the numbers out front) and subtract the exponents of matching bases. These rules let you simplify expressions quickly and form the foundation for working with polynomials and rational expressions later on.

Basics of Monomial Division

When you divide one monomial by another, you're looking for common factors you can cancel. Here's the general process:

1. Divide the coefficients (the numerical parts).
2. Identify matching variables in the numerator and denominator.
3. Subtract exponents for each matching base (numerator exponent minus denominator exponent).
4. Simplify any remaining terms, including negative exponents.

For example, take $\frac{15x^2y^3}{5xy^3}$. Both the top and bottom share the variables $x$ and $y$:

• Coefficients: $15 \div 5 = 3$
• For $x$: $x^{2-1} = x^1 = x$
• For $y$: $y^{3-3} = y^0 = 1$

Result: $3x$

Here's one that produces a fraction. Divide $\frac{18a^3b^2c}{6a^2bc^2}$:

• Coefficients: $18 \div 6 = 3$
• For $a$: $a^{3-2} = a$
• For $b$: $b^{2-1} = b$
• For $c$: $c^{1-2} = c^{-1} = \frac{1}{c}$

Result: $\frac{3ab}{c}$

Quotient Property of Exponents

The quotient property is the rule that makes all of this work. For any nonzero base $a$:

$\frac{a^m}{a^n} = a^{m-n}$

You keep the base and subtract the exponent in the denominator from the exponent in the numerator. A quick example:

$\frac{x^5}{x^2} = x^{5-2} = x^3$

For a full monomial like $\frac{12x^3y^2}{6x^2y}$, apply the property to each variable separately:

• $12 \div 6 = 2$
• $x^{3-2} = x$
• $y^{2-1} = y$

Result: $2xy$

Zero Exponents in Monomial Division

When the same variable has equal exponents in the numerator and denominator, the quotient property gives you an exponent of zero. The zero exponent rule says any nonzero base raised to the zero power equals 1:

$a^0 = 1 \quad (a \neq 0)$

So $5^0 = 1$ and $x^0 = 1$ (as long as $x \neq 0$).

This shows up naturally in division. Take $\frac{6x^3y^2}{2x^3y^2}$:

• $6 \div 2 = 3$
• $x^{3-3} = x^0 = 1$
• $y^{2-2} = y^0 = 1$

Result: $3 \cdot 1 \cdot 1 = 3$. The variables cancel completely, leaving just the coefficient.

Quotient to a Power Property

When an entire fraction is raised to a power, you can apply that power to the numerator and denominator separately:

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

For example:

$\left(\frac{x^2}{y^3}\right)^4 = \frac{(x^2)^4}{(y^3)^4} = \frac{x^8}{y^{12}}$

This property often appears alongside the quotient property. When it does, apply the power first, then simplify. Here's a more involved example:

1. Start with $\frac{(3x^2)^3}{(9x^5)^2}$
2. Apply the power to each factor: $\frac{3^3 \cdot x^6}{9^2 \cdot x^{10}} = \frac{27x^6}{81x^{10}}$
3. Divide coefficients: $27 \div 81 = \frac{1}{3}$
4. Subtract exponents: $x^{6-10} = x^{-4}$
5. Result: $\frac{1}{3x^4}$

Combining Multiple Exponent Properties

Harder problems require you to chain several properties together. The key is to work step by step and handle each piece separately.

Take $\frac{(2x^3y^2)^2(3xy^4)^3}{(6x^2y^3)^4}$:

1. Expand each power:

   • $(2x^3y^2)^2 = 4x^6y^4$
   • $(3xy^4)^3 = 27x^3y^{12}$
   • $(6x^2y^3)^4 = 1296x^8y^{12}$

2. Multiply the factors in the numerator: $4x^6y^4 \cdot 27x^3y^{12} = 108x^9y^{16}$

3. Divide: $\frac{108x^9y^{16}}{1296x^8y^{12}}$

   • $108 \div 1296 = \frac{1}{12}$
   • $x^{9-8} = x$
   • $y^{16-12} = y^4$

4. Result: $\frac{xy^4}{12}$

For problems with multiple variables, just apply the quotient property to each variable one at a time. For instance, $\frac{12x^3y^2z^4}{6x^2y^5z} = 2x^1y^{-3}z^3$, which simplifies to $\frac{2xz^3}{y^3}$ (see the next section for why).

Negative Exponents and Reciprocals

Division can produce negative exponents whenever the denominator's exponent is larger than the numerator's:

$\frac{x^2}{x^5} = x^{2-5} = x^{-3}$

A negative exponent means "take the reciprocal and make the exponent positive":

$x^{-3} = \frac{1}{x^3}$

To write a final answer without negative exponents, move any term with a negative exponent to the opposite part of the fraction:

• If it's in the numerator, move it to the denominator.
• If it's in the denominator, move it to the numerator.

For example, $2xy^{-3}z^3$ becomes $\frac{2xz^3}{y^3}$ because $y^{-3}$ flips down to the denominator as $y^3$.