Lesson 5: Divide Monomials

New Concept

Ready to master division with exponents? This lesson introduces the Quotient Property, showing how dividing like bases, such as $\frac{a^m}{a^n}$, simplifies to subtracting exponents. We'll also explore zero exponents and raising fractions to powers to divide monomials.

What’s next

This is just the start. Next, you'll tackle interactive examples and practice cards to master simplifying expressions and dividing monomials with confidence.

Property

If $a$ is a real number, $a \neq 0$, and $m$ and $n$ are whole numbers, then

Examples

• To simplify $\frac{x^8}{x^5}$, since $8 > 5$, we subtract the exponents: $x^{8-5} = x^3$.

• To simplify $\frac{y^4}{y^{10}}$, since $10 > 4$, the result is in the denominator: $\frac{1}{y^{10-4}} = \frac{1}{y^6}$.

Property

If $a$ is a non-zero number, then $a^0 = 1$. Any nonzero number raised to the zero power is 1. In this text, we assume any variable that we raise to the zero power is not zero.

Examples

• Any non-zero number to the zero power is 1, so $15^0 = 1$.

• For any non-zero variable $p$, it is true that $p^0 = 1$.

Property

If $a$ and $b$ are real numbers, $b \neq 0$, and $m$ is a counting number, then

To raise a fraction to a power, raise the numerator and denominator to that power.

Examples

• To simplify $(\frac{2}{5})^3$, we apply the exponent to both the numerator and the denominator: $\frac{2^3}{5^3} = \frac{8}{125}$.

• To simplify $(\frac{p}{4})^2$, we raise both the numerator and denominator to the second power: $\frac{p^2}{4^2} = \frac{p^2}{16}$.

Property

To divide monomials, rewrite the division as a fraction. Then, separate the expression into coefficients and variables with the same base. Simplify the coefficient fraction and use the Quotient Property for Exponents for each variable.

Examples

• Find the quotient $48y^9 \div 6y^2$. We rewrite this as $\frac{48y^9}{6y^2}$. Simplify the coefficients $\frac{48}{6}=8$ and the variables $\frac{y^9}{y^2}=y^{9-2}=y^7$. The result is $8y^7$.

• To simplify $\frac{36a^3b^5}{-9ab^2}$, we handle each part: $\frac{36}{-9} = -4$, $\frac{a^3}{a} = a^{3-1} = a^2$, and $\frac{b^5}{b^2} = b^{5-2} = b^3$. The result is $-4a^2b^3$.