Powers and Roots: Exponents & Square Roots
Exponents provide a shorthand for repeated multiplication — writing 5⁴ instead of 5 × 5 × 5 × 5 — while square roots reverse the process by asking "what number multiplied by itself gives this value?" Together they form the foundation for scientific notation, area and volume calculations, and the algebraic equations students will encounter next.
Components of Exponents & Square Roots
This section covers the key concepts:
- Understanding Exponents: The base is the number being multiplied, and the exponent tells how many times to multiply it by itself.
- Exponent Rules: Product rule (aᵐ × aⁿ = aᵐ⁺ⁿ), power rule ((aᵐ)ⁿ = aᵐˣⁿ), and zero exponent (a⁰ = 1).
- Square Roots: The square root of a number n is the value that, when multiplied by itself, equals n, written as √n.
- Perfect Squares & Estimation: Recognizing perfect squares (1, 4, 9, 16, 25, ...) and estimating non-perfect square roots.
Examples of Exponents & Square Roots
Understanding Exponents Examples
- Evaluate 2⁵: Multiply 2 × 2 × 2 × 2 × 2 = 32.
- Evaluate 10³: Multiply 10 × 10 × 10 = 1,000 — each power of 10 adds a zero.
- A bacteria colony doubles every hour. After 6 hours, the count is 2⁶ = 64 times the original.
Exponent Rules Examples
- Simplify 3² × 3⁴: Same base, so add exponents: 3²⁺⁴ = 3⁶ = 729.
- Simplify (2³)²: Multiply exponents: 2³ˣ² = 2⁶ = 64.
- Evaluate 7⁰: Any nonzero number raised to the zero power equals 1, so 7⁰ = 1.
Square Root Examples
- √49 = 7 because 7 × 7 = 49.
- √144 = 12 because 12 × 12 = 144.
- A square garden has an area of 81 square feet. Each side is √81 = 9 feet long.
Perfect Squares & Estimation Examples
- The first ten perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
- Estimate √50: Since 7² = 49 and 8² = 64, √50 is slightly more than 7 — approximately 7.07.
- Estimate √20: Since 4² = 16 and 5² = 25, √20 is between 4 and 5 — approximately 4.47.