Algebra Class: Exponents and Roots

Exponents, Roots, and Order of Operations

• Exponents: If a is a real number and n is a natural number, then aⁿ = a × a × ... × a (n factors of a). n is the exponent, a is the base.
• Evaluating Exponential Expressions:
  • 5² means 5 × 5 = 25
  • (0.2)³ means 0.2 × 0.2 × 0.2 = 0.008
• Sign of an Exponential Expression:
  • An even number of negative factors results in a positive product (e.g., (-2)(-2)(-2)(-2) = 16)
  • An odd number of negative factors results in a negative product (e.g., (-2)(-2)(-2)(-2)(-2) = -32)
• Square Roots:
  • The positive or principal square root of a number (written √a = b) is the positive number b such that b × b = a.
• Negative Square Root: The negative square root of a number, written −√a, is the negative of the positive square root
  • √25 = 5
  • −√25 = −5
• Square Roots of Negative Numbers: The square root of a negative number is not a real number.
• Order of Operations (PEMDAS/BODMAS): Follow these steps when evaluating expressions:
  • Step 1: Parentheses (or other grouping symbols)
  • Step 2: Exponents
  • Step 3: Multiplication and division (left to right)
  • Step 4: Addition and subtraction (left to right)

Evaluating Algebraic Expressions

• Algebraic expressions are combinations of numbers, variables, and operation symbols.
• To evaluate an algebraic expression, substitute given values for the variables.
• Use parentheses to avoid errors when substituting.

Linear Equations in One Variable

• A linear equation in one variable has the form ax + b = 0, where a and b are real numbers and a ≠ 0.
• A solution to an equation is a value that makes the equation true when substituted for the variable.
• Properties of equality: If a=b, then a+c=b+c and a×c=b×c (c≠0).

Solving Linear Equations

• Follow these steps:
  • Simplify both sides
  • Isolate variable terms (on one side)
  • Isolate the variable
  • Check the solution in the original equation

Solving a Linear Equation with Fractions or Decimals

• Multiply both sides by the least common denominator (LCD) of the fractions or decimals to eliminate fractions.

Identifying Conditional Equations, Contradictions, and Identities

• Conditional Equations: One solution (Example: 3x+2 = 8)
• Identities: Infinitely many solutions (Example: 5(x−3) = 5x - 15)
• Contradictions: No solution (Example: 2(x − 1) = 2x − 3)

Formulas and Percent

• Formulas use variables to represent relationships between quantities.
• Key: Treat the variable you want to solve for as the only one and treat other variables as numbers when solving for a specified variable.
• Percent: A ratio where the denominator is 100.
• Partial amount / whole amount = decimal value (converted to a percent)