Exponents and Polynomials: Rules and Applications

Exponents and Polynomials

The Quotient Rule for Exponents

The Quotient Rule for exponents is used when dividing exponential expressions with the same base. According to this rule, you subtract the exponent in the denominator from the exponent in the numerator.

• Definition: For any nonzero number a and integers m and n,

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

• Key Point: The bases must be the same to apply this rule.

• Zero Exponent Rule: Any nonzero number raised to the zero power is 1: $a^0 = 1$

• Example 1: $\frac{y^7}{y^5} = y^{7-5} = y^2$

• Example 2: $\frac{m^6}{m^6} = m^{6-6} = m^0 = 1$

• Example 3: $\frac{y^{27}}{y^9} = y^{27-9} = y^{18}$

• Example 4: $\frac{64}{16} = 4$

• Example 5: $\frac{12b^{11}}{4b^7} = 3b^{11-7} = 3b^4$

• Example 6: $\frac{9x^2y^8}{3xy^2} = 3x^{2-1}y^{8-2} = 3x^1y^6 = 3xy^6$

• Example 7: $\frac{30x^5y^3z^3}{-15x^2y^3z} = -2x^{5-2}y^{3-3}z^{3-1} = -2x^3z^2$

• Example 8: $\left(\frac{x^2}{x^3}\right)^3 = x^{2-3} = x^{-1}$, then $x^{-1} = \frac{1}{x}$

• Example 9: $\frac{9}{16} \div \frac{3}{4} = \frac{9}{16} \times \frac{4}{3} = \frac{9 \times 4}{16 \times 3} = \frac{36}{48} = \frac{3}{4}$

• Example 10: $\frac{3x^2y^8}{3xy^2} = x^{2-1}y^{8-2} = xy^6$

• Example 11: $\frac{(x^2)^4}{(x^3)^3} = \frac{x^{2 \times 4}}{x^{3 \times 3}} = \frac{x^8}{x^9} = x^{8-9} = x^{-1} = \frac{1}{x}$

• Example 12: $\frac{(3x^4)}{y^2}$ (expression for practice)

• Example 13: $\frac{5}{y}$ (expression for practice)

Summary Table: Exponent Rules

Name	Rule	Description
Product Rule	$a^m \times a^n = a^{m+n}$	Multiply terms with the same base: add exponents
Quotient Rule	$\frac{a^m}{a^n} = a^{m-n}$	Divide terms with the same base: subtract exponents
Zero Exponent Rule	$a^0 = 1$	Any nonzero base to the zero power is 1
Power Rule	$(a^m)^n = a^{mn}$	Raise a power to a power: multiply exponents
Power of a Product	$(ab)^m = a^m b^m$	Distribute exponent to each factor in parentheses
Power of a Quotient	$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$	Distribute exponent to numerator and denominator
Negative Exponent Rule	$a^{-n} = \frac{1}{a^n}$	Negative exponent means reciprocal