Exponents and Square Roots Study Guide for McGraw-Hill's Firefighter Exams

The exponent in an expression such as 3² shows how many times the base—in this case, 3—is multiplied by itself to find the result, known as the power.

Any base raised to the power of zero is equal to 1.

Examples: 60 = 1; 40 = 1. Any base raised to the power of 1 is equal to itself.

Example: 7¹ = 7; –55¹ = –55

A negative base number raised to an even exponent will be positive, while a negative base number raised to an odd exponent will be negative.

Examples: (–3)² = 9; (–3)³ = –27

Any base raised to a negative power is equal to the reciprocal of the base number (the number placed over 1) raised to the opposite power.

Example:

Multiplying Numbers with Exponents

When multiplying two like bases, each with a positive integer exponent, add the exponents.

Example: (4²) × (4³) = 4⁽²⁺³⁾ = 4⁵ = 1024

To verify this: (4² = 16) × (4³ = 64) = 16 × 64 = 1024.

When multiplying a base with two exponents, multiply the exponents.

Example: (23)4 = 2 (3×4) = 212 = 4096

Dividing Numbers with Exponents

When dividing two like bases, each with an exponent, subtract the exponents.

Example:

To verify this: (3³= 27) ÷ (3²= 9) = 3^(3-2) = 3¹ = 3

Square Roots and Radicals

Roots and exponents (powers) are reciprocal. The principal square root of a non-negative real number is denoted by the symbol Image where x represents the non-negative real number (x is greater than zero) whose square (the result of multiplying the number by itself) is x. The mark √ is called the radical sign and the number under the radical sign is called the radicand.

Examples:

The first ten square numbers and their roots are listed below.