O Level MathematicsC1.3 Powers and roots (squares, cubes, other integer powers/roots).

🔢 Power Up Your Math: Mastering Squares & Cubes!

Edudent Academy
26 Nov 25

Understanding powers and roots is **crucial** for success in O Level Mathematics. From simplifying algebraic expressions to solving equations and interpreting scientific notation, these concepts appear across many exam questions. Mastery here will boost both your speed and accuracy when tackling more advanced topics.

Mastering the Main Concepts

A power tells us how many times to multiply a number by itself. For example, 52=5×5=255^2 = 5\times5 = 25, while a root is the inverse operation: 25=5\sqrt{25} = 5. **Key idea:** squares (n2n^2) and cubes (n3n^3) are the most common, but you should also recognise higher integer powers/roots such as n4n^4 or n4\sqrt[4]{n}.

  • A positive number has two square roots: 16=4\sqrt{16}=4 and −16=−4-\sqrt{16}=-4.
  • The cube root of any real number is unique: −273=−3\sqrt[3]{-27}=-3.
  • Powers distribute over multiplication: (ab)n=anbn(ab)^n = a^n b^n.
  • To simplify fractions: (ab)n=anbn\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}.

Problem: Calculate (a) 144\sqrt{144}, (b) 333^3, and (c) 814\sqrt[4]{81}.\nStep 1: Identify each operation. Part (a) is a square root, part (b) is a cube (power of 3), and part (c) is a 4th root.\nStep 2: Evaluate each one. (a) 144=12\sqrt{144}=12 because 122=14412^2=144. (b) 33=3×3×3=273^3 = 3\times3\times3 = 27. (c) 814=3\sqrt[4]{81}=3 because 34=813^4 = 81.\nStep 3: State the answers clearly: 12, 27, and 3.\nWith regular practice, these steps become automatic—so keep working through textbook questions and past papers!