Powers represent repeated multiplication. For example, 2⁴ means 2 × 2 × 2 × 2 = 16. The small raised number is called the exponent, and it tells you how many times to multiply the base (the larger number) by itself. Powers are a shorthand that makes it easier to write and work with very large or very small numbers.
Exponent laws are rules that help you simplify expressions with powers. Here are the key laws: Product rule — when multiplying powers with the same base, add the exponents (aᵐ · aⁿ = aᵐ⁺ⁿ). Quotient rule — when dividing powers with the same base, subtract the exponents (aᵐ ÷ aⁿ = aᵐ⁻ⁿ). Power of a power — when raising a power to another power, multiply the exponents ((aᵐ)ⁿ = aᵐⁿ). Power of a product — distribute the exponent to each factor ((ab)ⁿ = aⁿ · bⁿ). Zero exponent — any non-zero number to the power of 0 equals 1 (a⁰ = 1). These laws make calculations faster and expressions cleaner.
Roots are the inverse operation of powers. The square root of a number is the value that, when squared, gives you that number. For example, √49 = 7 because 7² = 49. Perfect squares are numbers like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on — they're the result of squaring whole numbers. When you need to find the square root of a non-perfect square like √50, you can estimate by finding the two perfect squares it falls between (49 and 64), so √50 is between 7 and 8, closer to 7.
Scientific notation is a way to write very large or very small numbers compactly. It has the form a × 10ᵏ, where 1 ≤ a < 10 and k is an integer. For example, 4,500,000 = 4.5 × 10⁶ (move the decimal 6 places left, so the exponent is +6). For small numbers like 0.00032, write it as 3.2 × 10⁻⁴ (move the decimal 4 places right to get 3.2, so the exponent is −4). When multiplying or dividing in scientific notation, work with the coefficients separately and apply exponent laws to the powers of 10.
Real-world connection: Scientific notation is used in astronomy (distances between stars), chemistry (size of atoms), and computing (data storage capacities). Exponent laws simplify calculations in physics and engineering.
Exponent laws — examples with base 2
| Expression | Rule | Calculation | Result |
|---|---|---|---|
| 2³ · 2⁴ | Product (add exponents) | 2³⁺⁴ = 2⁷ | 128 |
| 2⁸ ÷ 2⁵ | Quotient (subtract exponents) | 2⁸⁻⁵ = 2³ | 8 |
| (2³)² | Power of power (multiply) | 2³ˣ² = 2⁶ | 64 |
| 2⁰ | Zero exponent | Any base⁰ = 1 | 1 |
Perfect squares and square roots
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| n² | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 |
| √(n²) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Scientific notation conversion
| Standard Form | Scientific Notation | Exponent Sign |
|---|---|---|
| 4,500,000 | 4.5 × 10⁶ | Positive (large number) |
| 0.00032 | 3.2 × 10⁻⁴ | Negative (small number) |
| 780 | 7.8 × 10² | Positive |
| 0.0056 | 5.6 × 10⁻³ | Negative |