Roots and Perfect Numbers

Overview

This lecture covers how to evaluate square roots and cube roots, work with perfect squares and cubes, estimate roots, and compare real numbers.

The square root of a number p is a value x such that x² = p.
The cube root of a number p is a value x such that x³ = p.
Square roots and cube roots can have both positive and negative solutions, e.g., ±9 for √81.
The radical symbol (√ for square root, ∛ for cube root) is used to indicate roots.
For fractions, take the root of the numerator and denominator separately, e.g., √(4/9) = 2/3.

Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256 (up to 16²).
Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 (up to 10³).
Memorizing perfect squares and cubes helps estimate non-perfect roots.

The root of a non-perfect square or cube is irrational and cannot be exactly written as a fraction or finite decimal.
Use root symbols (√ or ∛) to represent irrational roots, e.g., √2 or ∛10.

To estimate √50: find perfect squares around 50 (49 and 64), so √50 is between 7 and 8, approximately 7.1.
To estimate ∛100: find perfect cubes around 100 (64 and 125), so ∛100 is between 4 and 5, approximately 4.6.
The closer the number is to one perfect root, the closer its estimate.

To compare values involving roots, estimate them and use < or > accordingly.
Examples:
- √3 + 2 < √3 + 6
- -1 > -3, so √7 - 1 > √7 - 3
- 2 × √6 < 2 × √7
- -5 × √3 > -7 × √3

Square Root — A number which, when squared, gives the original number (x where x² = p).
Cube Root — A number which, when cubed, gives the original number (x where x³ = p).
Radical Symbol — The notation (√ or ∛) indicating a root.
Perfect Square — A whole number whose square root is an integer.
Perfect Cube — A whole number whose cube root is an integer.
Irrational Number — Cannot be written as a fraction; non-repeating, non-terminating decimal.
Real Numbers — All numbers on the number line, including rational and irrational numbers.