Root

A nth root, r, of a number m, is a number such that rⁿ = m. This is different from the root of a function – e.g. a root of a quadratic.

The root students most frequently come across, the square root (or 2nd root) r, of a number m, is a number such that r² = m. For example, a square root of 25 is 5 because 5² = 5 × 5 = 25. Note that −5 is another square root of 25, because (−5)² = 25 also.

There is a mathematical symbol used for positive square roots:

\(\sqrt{25}=5\) which can be read as “the (positive) square root of 25 is equal to 5.”

In simpler terms: the square root of a number m, is the number that when squared (multiplied by itself), gets you the original number m.

The cube root (or 3rd root) r, of a number m, is a number such that r³ = m. For example, a cube root of 64 is 4 because 4³ = 4 x 4 x 4 = 64. The mathematical symbol for higher roots is similar to the square root symbol – but a number is added to the outside to show the type of root. For example, we could write \(\sqrt[3]{64}=4\).

nth roots of integers which are not perfect powers of n are surds.

Relevant lessons:

• N3b – Order of operations, including brackets, powers, roots and reciprocals
• N6a – Using powers and roots
• N6b – Estimating powers and roots
• N7a – Calculating with roots and with integer indices
• N7b – Calculating with fractional indices
• N8b – Surds