Squares, Square Roots, and Exponent Notation

Squares and Square Roots

• Squaring a number involves multiplying it by itself, denoted as 𝑛².
• Example: 5² = 5 × 5 = 25.
• The square of any integer is a non-negative number, called a perfect square.
• Examples of perfect squares: 1, 4, 9, 16, 25.
• Perfect squares have integer square roots.
• A square root is the inverse operation of squaring.
• If x is the square of n, then n is the square root of x, represented as √𝑥.
• Example: √25 = 5 because 5 × 5 = 25.
• Square roots of non-perfect squares result in irrational numbers (e.g., √2).
• Negative numbers do not have real square roots.

Exponent Notation

• Exponent notation is a shorthand for repeated multiplication, written as 𝑎ⁿ.
• 𝑎 is the base, and 𝑛 is the exponent or power.
• Example: 3⁴ = 3 × 3 × 3 × 3 = 81.
• Product of Powers: 𝑎ᵐ × 𝑎ⁿ = 𝑎ᵐ⁺ⁿ (add exponents when multiplying numbers with the same base).
• Power of a Power: (𝑎ᵐ)ⁿ = 𝑎ᵐ×ⁿ (multiply exponents when raising a power to another power).
• Power of a Product: (𝑎𝑏)ⁿ = 𝑎ⁿ × 𝑏ⁿ (distribute the exponent across the product).
• Power of a Fraction: (𝑎/𝑏)ⁿ = 𝑎ⁿ/𝑏ⁿ (exponent applies to both numerator and denominator).
• Zero Exponent: 𝑎⁰ = 1 (any non-zero number raised to the power of zero equals 1).
• Negative Exponent: 𝑎⁻ⁿ = 1/𝑎ⁿ (negative exponent represents the reciprocal of the base raised to the positive exponent).

Prime Factor Form (and Factor Trees)

• Prime factorization breaks down a number into its prime factors.
• A prime number is greater than 1 and divisible only by 1 and itself.
• Example: The prime factorization of 36 is 2² × 3².
• A factor tree is a visual tool to find the prime factorization.
• Start by dividing the number by its smallest prime factor and continue dividing each quotient.
• Example: Factor tree for 36: 36 → 2 × 18 → 2 × 2 × 9 → 2 × 2 × 3 × 3.
• Prime factorization is used to find the greatest common divisor (GCD) and the least common multiple (LCM).

Common Factors (HCF)

• The Highest Common Factor (HCF), also called the Greatest Common Divisor (GCD), is the largest number that divides two or more numbers exactly.
• To find the HCF of 24 and 36, list the factors of each number.
• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• The common factors are 1, 2, 3, 4, 6, and 12, so the HCF is 12.
• Using prime factorization:
• 24 = 2³ × 3
• 36 = 2² × 3²
• HCF is the product of the lowest powers of all common prime factors, so the HCF is 2² × 3 = 12.
• The HCF simplifies fractions and solves problems involving divisibility and sharing.

Common Multiples (LCM)

• The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers.
• The LCM of 4 and 5 is 20.
• List the multiples of each number and identify the smallest common one.
• Multiples of 4: 4, 8, 12, 16, 20, 24,...
• Multiples of 5: 5, 10, 15, 20, 25,... The LCM is 20.
• Using prime factorization:
• 4 = 2²
• 5 = 5¹
• LCM = 2² × 5 = 20
• The LCM solves problems involving synchronization.

Expanded Notation

• Expanded notation shows the value of each digit based on its place value.
• Example: 3456 = 3000 + 400 + 50 + 6.
• Scientific notation expresses numbers as a product of a coefficient between 1 and 10, and a power of 10.
• Example: 5000 = 5 × 10³.
• Decimal notation deals with fractional parts.
• Example: 0.045 = 4 × 10⁻² + 5 × 10⁻³.

Learn about squaring numbers, perfect squares, and square roots. Understand exponent notation as shorthand for repeated multiplication. Explore rules like product of powers and power of a power.