Algebra 1 Unit 1: Weeks 3-4 Flashcards

Properties of Exponents

• Product of Powers Rule: When multiplying exponents with the same base, add the exponents (e.g., (b^3 \cdot b^2 = b^{3+2} = b^5)).

• Quotient of Powers Property: When dividing exponents with the same base, subtract the exponents (e.g., (\frac{x^6}{x^5} = x^{6-5} = x^1)).

• Power of a Power Property: Raising a power to another power involves multiplying the exponents (e.g., ((a^m)^n = a^{m \cdot n})).

• Power of a Product Property: To find a power of a product, apply the exponent to each factor individually and multiply (e.g., ((xb)^c = x^c \cdot b^c)).

• Power of a Quotient Property: When calculating the power of a quotient, find the power of both the numerator and the denominator, then divide them (e.g., ((\frac{1}{2})^2 = \frac{1^2}{2^2})).

Negative Exponents and Reciprocals

• Negative Exponent Rule: Move a base with a negative exponent from the numerator to the denominator and convert the exponent to positive (e.g., (a^{-n} = \frac{1}{a^n})).

• Reciprocal of Negative Exponent: The reciprocal of (b^{-1}) is (1/b^1) which simplifies to (1/b).

• Reciprocal of Positive Exponent: The reciprocal of (b^1) is (1/b^{-1}).

Special Cases and Simplifications

• Any Number to the Zeroth Power: Any non-zero number raised to the zero power equals one, including negative numbers (e.g., (x^0 = 1)).

• Order of Operations (GEMDAS): The correct order for solving expressions is Grouping, Exponents, Multiplication, Division, Addition, and Subtraction.

• Simplifying Expressions: For example, (-(3x)^2) simplifies to (-9x^2).

Truth Statements and Expressions

• Verification of Expressions: The statement ( \frac{a^{-3}}{a^{-4}} = \frac{a}{1} ) is FALSE.

• Example of Power Calculation: (((x + 2)^2)^3) simplifies to ((x + 2)^{2 \cdot 3} = (x + 2)^6).

• Fraction Formality: An example of converting to fractional form is (3^{-1} = \frac{1}{3}).