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)

Test your knowledge on exponents, roots, and the order of operations in this algebra quiz. Learn how to evaluate exponential expressions, understand the signs of products, and tackle square roots. Perfect for reinforcing fundamental algebra concepts.