Cube and Cube Roots in Mathematics

Cube and Cube Roots

• The cube of a number is found by multiplying the number by itself three times, e.g., (4 \times 4 \times 4 = 64).
• The process of finding a cube root identifies a number which, when cubed, yields the original number; for instance, (4) is the cube root of (64) since (64 = 4^3).

Definition of Cube Root

• The cube root of a number "a" is the value that, when cubed, equals "a".
• The symbol ( \sqrt[3]{a} ) represents the cube root of "a".

Properties of Cube Roots

• The cube root of a positive number is also positive.
• The cube root of a negative number is negative, preserving the sign of the original number (e.g., ( \sqrt[3]{64} = 4) and ( \sqrt[3]{-27} = -3)).

Perfect Cube Rational Numbers

• A perfect cube rational number can be expressed as the cube of a rational number.
• For example, (27 = (3)^3) and (-8 = (-2)^3).
• The cube root of a perfect cube rational number is also a rational number.

Non-Perfect Cubes

• If a number is not a perfect cube, its cube root is denoted using the cube root symbol, ( \sqrt[3]{x} ).