Powers and Roots: Bases, Exponents and Radicals Demystified

A power is written as aⁿ, where 'a' is the base and 'n' is the exponent. The exponent indicates how many times the base is used as a factor in a multiplication. So 2³ = 2 × 2 × 2 = 8. We read this as 'two cubed' or 'two to the power of three'. The result, 8, is called the power or value of the expression.

Special cases of the exponent deserve attention. Any number (except zero) raised to the exponent 1 is itself: 5¹ = 5. Any number (except zero) raised to the exponent 0 is always 1: 7⁰ = 1, 100⁰ = 1. This may seem arbitrary, but there is a reason: dividing aⁿ by a lowers the exponent by 1. If aⁿ ÷ a = aⁿ⁻¹ and we reach a¹ ÷ a = a⁰, the result must be 1.

Negative exponents represent the reciprocal. a⁻ⁿ = 1/aⁿ. Therefore 2⁻³ = 1/2³ = 1/8 = 0.125. This is very useful in scientific notation and in formulas from physics and chemistry, where negative powers of 10 represent very small values, such as 10⁻⁶ metres (one micrometre).

The product rule: when multiplying powers with the same base, add the exponents. aᵐ × aⁿ = aᵐ⁺ⁿ. For example, 3² × 3⁴ = 3⁶ = 729. This makes sense because 3² = 3 × 3 and 3⁴ = 3 × 3 × 3 × 3, so together they are six factors of 3.

The quotient rule: when dividing powers with the same base, subtract the exponents. aᵐ ÷ aⁿ = aᵐ⁻ⁿ. So 5⁵ ÷ 5² = 5³ = 125. This rule also explains why a⁰ = 1: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, but any number divided by itself is 1.

The power of a power rule: (aᵐ)ⁿ = aᵐ × ⁿ. For example, (2³)² = 2⁶ = 64. And the product raised to a power: (a × b)ⁿ = aⁿ × bⁿ. So (2 × 3)⁴ = 2⁴ × 3⁴ = 16 × 81 = 1296. These properties reduce complex expressions to simple calculations.

Radicals are the inverse of exponentiation. The square root of a number n is the value that, when squared, equals n. √25 = 5 because 5² = 25. The cube root of n is the value that, when cubed, equals n: ∛27 = 3 because 3³ = 27.

Roots of perfect squares or cubes have exact results: √4 = 2, √9 = 3, √16 = 4, √36 = 6, √100 = 10. Roots of non-perfect numbers are irrational: √2 ≈ 1.41421…, √3 ≈ 1.73205…, √5 ≈ 2.23606… These numbers have infinitely many decimal places with no repeating pattern — they cannot be written as a fraction of two integers.

The relationship between roots and fractional exponents is fundamental: √a = a^(1/2) and ∛a = a^(1/3). This means the exponent rules apply to radicals as well. For example, √(a × b) = √a × √b, which allows radical simplification: √12 = √(4 × 3) = √4 × √3 = 2√3.

Simplifying a radical means extracting perfect factors from under the radical sign. For √72: factorise 72 = 36 × 2, where 36 is a perfect square. Then √72 = √36 × √2 = 6√2. For √180: 180 = 4 × 9 × 5 = 36 × 5. So √180 = 6√5.

The most familiar irrational numbers in secondary school are √2, √3, √5 and π. The number √2 appears as the diagonal of a unit square (Pythagorean theorem: 1² + 1² = 2, so the diagonal is √2). The irrationality of √2 shocked the ancient Greek mathematicians, who believed every number could be expressed as a ratio of integers.

Operations with radicals require care: you can only add and subtract like radicals (same root index and same radicand). 3√2 + 5√2 = 8√2, just as 3x + 5x = 8x. But √3 + √5 cannot be simplified — they are unlike radicals.

Scientific notation uses powers of 10 to represent very large or very small numbers compactly. A number in scientific notation is written as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. The distance from Earth to the Sun is approximately 1.5 × 10¹¹ metres. The diameter of a hydrogen atom is about 5.3 × 10⁻¹¹ metres.

To convert to scientific notation: move the decimal point until only one non-zero digit remains before it, and count how many places you moved. If you moved left, the exponent is positive; if right, it is negative. Example: 45,000 → move 4 places left → 4.5 × 10⁴. Example: 0.0037 → move 3 places right → 3.7 × 10⁻³.

In area calculations, exponentiation is indispensable. The area of a square with side L is L². The area of a circle with radius r is πr². If the side doubles, the area quadruples — because (2L)² = 4L². This explains why small changes in dimensions have a large impact on areas and volumes, something very relevant in engineering, architecture and physics.