Fundamentals of Algebra: Concepts & Practice

1.1 Exponents

• An exponent tells how many times to multiply a number by itself. Example: \(2^3 = 2 \times 2 \times 2 = 8\).

1.2 Scientific Notation

• Numbers written as \(a \times 10^n\), where \(1 \leq a < 10\) and \(n\) is an integer. Example: \(4,500 = 4.5 \times 10^3\).

1.3 Order of Operations

• Sequence: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS).

1.4 Commutative & Associative Properties

• Commutative: Order doesn't matter (e.g., \(a + b = b + a\)).
• Associative: Grouping doesn't matter (e.g., \((a + b) + c = a + (b + c)\)).

1.5 Distributive & Identity Properties

• Distributive: \(a(b + c) = ab + ac\).
• Identity: Adding 0 or multiplying by 1 does not change a number.

1.6 Zero & Equality Properties

• Zero Property: Any number times 0 is 0.
• Equality Properties: If \(a = b\), then \(a + c = b + c\), etc.

1.7 Factors & Multiples

• Factors: Numbers that divide exactly into another number.
• Multiples: Products of a number and an integer.

1.8 Variable Expressions & Equations

• Variable Expression: Contains numbers, variables, and operations (e.g., \(3x + 2\)).
• Equation: States that two expressions are equal (e.g., \(x + 4 = 10\)).

1.9 Solving Equations & Inequalities

• To solve, use inverse operations to isolate the variable.
• Inequality: Compares values (e.g., \(x > 5\)).
• Substitution: Plugging a value into a variable to test or solve.

• Exponents: \(5^2 = 25\)
• Scientific Notation: \(0.0008 = 8 \times 10^{-4}\)
• Order of Operations: \(6 + 2 \times 3^2 = 6 + 2 \times 9 = 6 + 18 = 24\)
• Commutative: \(7 + 2 = 2 + 7\)
• Associative: \((1 + 2) + 3 = 1 + (2 + 3)\)
• Distributive: \(3(x + 4) = 3x + 12\)
• Identity: \(9 \times 1 = 9\)
• Zero Property: \(17 \times 0 = 0\)
• Factors: Factors of 12: 1, 2, 3, 4, 6, 12
• Multiples: Multiples of 4: 4, 8, 12, 16, ...
• Variable Expression: If \(x = 5\), then \(2x + 3 = 2 \times 5 + 3 = 13\)
• Solving by Addition: \(x - 5 = 7\) ⇒ \(x = 12\)
• Solving by Multiplication: \(\frac{x}{3} = 4\) ⇒ \(x = 12\)
• Inequality: \(x + 2 > 5\) ⇒ \(x > 3\)
• Substitution: If \(y = 3\), in \(2y + 1\) ⇒ \(2 \times 3 + 1 = 7\)

1. Q: What is an exponent?
   A: It shows how many times to multiply a number by itself.
2. Q: How do you write 5,200 in scientific notation?
   A: \(5.2 \times 10^3\)
3. Q: What is PEMDAS?
   A: The order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
4. Q: What is the commutative property?
   A: The order of addition or multiplication does not affect the result.
5. Q: What is the distributive property?
   A: \(a(b + c) = ab + ac\)
6. Q: What is the zero property?
   A: Any number multiplied by 0 is 0.
7. Q: What is a factor?
   A: A number that divides evenly into another number.
8. Q: What is a multiple?
   A: A product of a number and any integer.
9. Q: What is a variable expression?
   A: An expression containing variables, numbers, and operations.
10. Q: How do you solve \(x + 3 = 7\)?
    A: Subtract 3 from both sides: \(x = 4\).