Radical

High School

Definition

What is referred to when you see the square root sign (√). Radical expressions have at least one number under a radical. The sign can mean square root, but it can also mean some other root. Radicals that aren't square roots will have a smaller number (a subscript) that allows you to identify which root you're dealing with.

Worked examples

\(\sqrt{25} = 5\)

The radical sign without a subscript means square root — what number squared gives 25?

\(\sqrt[3]{8} = 2\)

The small 3 makes this a cube root — what number cubed gives 8?

\(\sqrt[4]{16} = 2\)

A subscript of 4 means fourth root — what number to the fourth power gives 16?

Common mistakes

\(\sqrt{25} = \pm 5\) → \(\sqrt{25} = 5\) The radical symbol means the principal (non-negative) root only; both solutions appear when solving \(x^2 = 25\).
\(\sqrt[3]{8}\) written as \(\sqrt{8}\) → \(\sqrt[3]{8}\) Omitting the subscript changes it to a square root instead of a cube root.
\(\sqrt{{-9}} = -3\) → \(\sqrt{{-9}}\) is not a real number Square roots of negative numbers are undefined in the real number system (you'll meet them later as imaginary numbers).

Where you'll use it next

You'll use radicals to simplify expressions, solve radical equations, work with exponent rules (fractional exponents), and later in geometry for distance formulas and trigonometry.

Found in 1 StudyPug lesson

Simplifying Radicals: A Comprehensive Guide

Precalculus

Unlock the secrets of simplifying radicals with our step-by-step approach. Learn to tackle square roots, cube roots, and complex expressions with confidence and precision.

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