Expressions & Equations

Square & Cube Roots

Using square root and cube root symbols to represent solutions to equations.

Intermediate15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Properties of Exponents

Undoing a Power

Squaring 7 gives 49. Before reading on, think about the reverse question: what number, squared, gives 49? Is there only one answer, or more than one?

Definition — Square Root and Cube Root

A square root of a number is a value that, multiplied by itself, gives that number — written √a. A cube root is a value that, multiplied by itself three times, gives that number — written ∛a. The radical symbol √ by itself denotes the non-negative (principal) square root.

Both 7 and −7 square to 49, since a negative times a negative is positive — so the equation x² = 49 truly has two solutions, x = 7 and x = −7. But the radical symbol √49 by itself is defined to mean only the positive one, 7, so that √ always names one specific, unambiguous value rather than a pair. Cube roots don't have this ambiguity: only one real number, cubed, gives any particular result, including negative results — (−3)³ = −27, so ∛(−27) = −3 with no second answer to worry about.

Worked Example — Solving an Equation with a Squared Variable

Solve x² = 36. Both the positive and negative square roots of 36 satisfy this equation: x = 6 or x = −6, often written x = ±6.

Worked Example — Evaluating Cube Roots

Evaluate ∛64 and ∛(−8). Since 4³ = 64, ∛64 = 4. Since (−2)³ = −8, ∛(−8) = −2 — a cube root of a negative number is a perfectly ordinary negative number, unlike a square root of a negative number.

x^{2} = 49 \;\Longrightarrow\; x = \pm 7 \qquad \sqrt[3]{-27} = -3

Tip

Use ± only when solving an equation like x² = a for every possible value of x — the radical symbol √a by itself always means just the single, non-negative principal root.

Common Mistakes

Writing √49 as ±7, treating the radical symbol itself as if it already includes both signs.
The radical symbol √ always denotes only the non-negative principal root — √49 = 7 exactly, with the ± only appearing when solving an equation like x² = 49 for every solution.
Assuming a cube root of a negative number is undefined, by analogy with square roots of negative numbers.
Cube roots of negative numbers are perfectly well-defined real numbers — since a negative number cubed stays negative, every real number has exactly one real cube root, positive or negative.

Key Takeaways

A square root undoes squaring; a cube root undoes cubing.
The equation x² = a has two solutions, ±√a, but the radical symbol √a alone means only the non-negative root.
Every real number, including negative numbers, has exactly one real cube root.

Summary

Square and cube roots undo exponents, with an important distinction in how many solutions each type of root equation has. The next lesson uses exponents for a very different purpose — writing extremely large or small numbers compactly.

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