7.5 Square Roots and Cube Roots

Square Roots and Cube Roots examines the specific behavior of the two most commonly used roots — how each is defined for perfect powers, zero, positive non-perfect values, and negative radicands — and the distinction between a principal root and the full set of solutions to a power equation.

Square Roots

Principal Square Root

Every positive number has two square roots, one positive and one negative, but the radical symbol by convention refers only to the principal square root, which is the nonnegative one. When the radical symbol is used, it always denotes this single nonnegative value, even though a corresponding negative root also exists.

Square Root of a Perfect Square

The square root of a perfect square is a whole number, found by identifying which whole number, when multiplied by itself, produces the given radicand exactly.

Square Root of Zero

The square root of zero is zero, since zero multiplied by itself produces zero, and this is the unique case in which the positive and negative square roots coincide at a single value.

Square Root of a Positive Nonperfect Square

The square root of a positive number that is not a perfect square is an irrational number, whose decimal expansion continues without terminating or repeating, and such a root can only be expressed exactly using the radical symbol itself or approximated to a chosen number of decimal places.

Negative Radicand Square Root in the Real Numbers

The square root of a negative number is not defined within the real numbers, since no real number multiplied by itself produces a negative result — squaring a positive or a negative real number always yields a nonnegative value.

Cube Roots

Cube Root of a Perfect Cube

The cube root of a perfect cube is a whole number, found by identifying which whole number, when multiplied by itself twice more, produces the given radicand exactly. Unlike the square root, the cube root does not require distinguishing a principal value from a negative counterpart, since every real number has exactly one real cube root.

Cube Root of Zero

The cube root of zero is zero, since zero multiplied by itself any number of times remains zero.

Cube Root of a Negative Number

The cube root of a negative number is defined and is itself negative, since a negative number multiplied by itself twice more produces a negative result. This makes cube roots defined for all real numbers, in contrast to square roots, which are undefined for negative radicands within the real numbers.

Verifying and Distinguishing Roots

Power and Root Verification

A computed root can be verified by raising the result back to the corresponding power and confirming that it reproduces the original radicand; if squaring the claimed square root, or cubing the claimed cube root, does not return the original value, the root was computed incorrectly.

Principal Root and Equation Solution Distinction

The principal square root notation must be distinguished from the full solution set of an equation such as x squared equals a given value. While the radical symbol always returns only the single nonnegative principal root, solving an equation of that form yields both the positive and negative roots as valid solutions, since both values satisfy the squared equation.