Cubes and Cube Roots - Finding Cube Roots by Prime Factorisation

Definition of a Cube: A number $x$ is said to be the cube of a number $n$ if $x = n \times n \times n$. Geometrically, this represents the volume of a solid cube with side length $n$, where all three dimensions (length, breadth, and height) are equal.

Perfect Cubes: A natural number is called a perfect cube if it is the cube of some natural number. When expressed as a product of prime factors, every prime factor of a perfect cube appears in groups of three (triplets). Visually, think of a prime factor tree where every leaf node can be neatly bundled into sets of three identical numbers.

Properties of Cubes: The cubes of all even natural numbers are even (e.g., $2^3 = 8$, $4^3 = 64$), and the cubes of all odd natural numbers are odd (e.g., $3^3 = 27$, $5^3 = 125$).

Cube Root Notation: The cube root of a number $x$ is the number $n$ such that $n^3 = x$. It is denoted by the radical symbol $\sqrt[3]{x}$. For example, since $4^3 = 64$, we write $\sqrt[3]{64} = 4$.

Finding Cube Root by Prime Factorisation: This method involves three main steps: (1) Resolve the given number into its prime factors. (2) Group the identical factors into triplets. (3) Take one factor from each triplet and find their product to obtain the cube root. Visually, this is like decomposing a large block into smaller equal units.

Cube Roots of Negative Numbers: The cube root of a negative perfect cube is always negative. For any positive integer $x$, $\sqrt[3]{-x} = -\sqrt[3]{x}$. On a number line, this signifies moving to the left of zero by the same magnitude as the positive root.

Cube Roots of Product and Fractions: The cube root of a product of two numbers is equal to the product of their individual cube roots: $\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}$. Similarly, for fractions, $\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$, which allows for simplifying complex ratios step-by-step.