Exponents and Polynomials: Rules, Properties, and Applications

Ch. 4 - Exponents and Polynomials

Product Rule for Exponents

The Product Rule for Exponents is a fundamental property used to simplify expressions involving exponents. When multiplying terms with the same base, you add their exponents.

• Definition: For any real number a and integers m and n, $a^m \times a^n = a^{m+n}$

• Key Point: The base must be the same for the rule to apply.

• Example: $x^{30} \cdot x^{70} = x^{100}$

• Example: $a^m \times a^n = a^{m+n}$

• Example: $(2s^3t)(3s^4t^2) = 6s^{3+4}t^{1+2} = 6s^7t^3$

• Example: $(-5a^2)(3a^8) = -15a^{2+8} = -15a^{10}$

• Example: $xy \cdot 3x^2 = 3x^{1+2}y = 3x^3y$

Quotient Rule for Exponents

The Quotient Rule for Exponents is used when dividing terms with the same base. Subtract the exponent in the denominator from the exponent in the numerator.

• Definition: For any real number a (except 0) and integers m and n, $\frac{a^m}{a^n} = a^{m-n}$

• Key Point: The base must be the same for the rule to apply.

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

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

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

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

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

• General Rule: $\frac{a^m}{a^n} = a^{m-n}$

Zero Exponent Rule

The Zero Exponent Rule states that any nonzero base raised to the zero power is equal to 1.

• Definition: For any nonzero real number a, $a^0 = 1$

• Example: $(-4t)^0 = 1$ (for $t \neq 0$)

• Example: $a^0 = 1$

Power Rule for Exponents

The Power Rule for Exponents is used when raising a power to another power. Multiply the exponents.

• Definition: For any real number a and integers m and n, $(a^m)^n = a^{m \cdot n}$

• Example: $(((-2)^3)^5) = (-2)^{3 \cdot 5} = (-2)^{15}$

• Example: $(y^8)^4 = y^{8 \cdot 4} = y^{32}$

• General Rule: $(a^m)^n = a^{m \cdot n}$

Power of a Product Rule

The Power of a Product Rule (also called the product to a power rule) allows you to distribute an exponent to each factor inside parentheses.

• Definition: For any real numbers a and b, and integer m, $(ab)^m = a^m b^m$

• Example: $(3 \cdot 4)^2 = 3^2 \cdot 4^2 = 9 \cdot 16 = 144$

• Example: $(5 \cdot 3^2)^2 = 5^2 \cdot (3^2)^2 = 25 \cdot 81 = 2025$

Simplifying Exponential Expressions Using All Exponent Rules

To fully simplify exponential expressions, you may need to use multiple exponent rules. The typical process involves working from the innermost expressions outward and applying the following checklist:

• No powers raised to other powers (apply power rules)

• No parentheses (apply power of a product rule)

• No same bases multiplied or divided (apply product and quotient rules)

• No zero exponents (apply zero exponent rule)

• No negative exponents (apply negative exponent rule: $a^{-n} = \frac{1}{a^n}$)

• All numbers with exponents evaluated

• All operations (multiplication, division, addition, subtraction) performed

Example: Simplify $(3x^{-5})^2 \cdot (-2x^4)^3$

Example: Simplify $\left(\frac{x^2y^7}{x^5y^4}\right)^{-1}$

Polynomials: Definitions and Classification

Introduction to Polynomials

A polynomial is an algebraic expression where variables have only whole number exponents (no negatives, no fractions). Polynomials are classified by the number of terms:

• Monomial: One term (e.g., $5x^3$)

• Binomial: Two terms (e.g., $x^2 + 3x$)

• Trinomial: Three terms (e.g., $x^2 + 3x + 2$)

To determine if an expression is a polynomial, check for whole number exponents and count the number of terms.

Multiplying Polynomials

Multiplying Polynomials by Monomials

To multiply a polynomial by a monomial (single term), use the distributive property:

• Example: $3x^2(y + 2x) = 3x^2y + 6x^3$

Multiplying Binomials Using the FOIL Method

When multiplying two binomials, use the FOIL method (First, Outer, Inner, Last) to ensure all terms are multiplied correctly:

• First: Multiply the first terms in each binomial.

• Outer: Multiply the outer terms.

• Inner: Multiply the inner terms.

• Last: Multiply the last terms.

Example: $(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$

Multiplying Polynomials with More Than Two Terms

For polynomials with more than two terms, distribute each term of the shorter expression to every term of the longer expression, then combine like terms.

• Example: $(x + 2)(x^2 + 3x + 4) = x(x^2 + 3x + 4) + 2(x^2 + 3x + 4)$

Summary Table: Exponent Rules

Name	Rule	Description
Product Rule	$a^m \times a^n = a^{m+n}$	Multiply terms with same base: add exponents
Quotient Rule	$\frac{a^m}{a^n} = a^{m-n}$	Divide terms with same base: subtract exponents
Zero Exponent Rule	$a^0 = 1$	Any nonzero base to zero power is 1
Power Rule	$(a^m)^n = a^{m \cdot n}$	Power to another power: multiply exponents
Power of a Product Rule	$(ab)^m = a^m b^m$	Distribute exponent to each factor in parentheses