7.4 Root Notation and Meaning

Root Notation and Meaning describes the radical symbol used to express the inverse operation of raising a number to a power, including the roles of the index and radicand, the conventions for reading square, cube, and higher roots, and the distinction between exact and approximate root values.

Roots as the Inverse of Powers

Root as an Inverse Power Relationship

A root answers the question of what number, when raised to a given power, produces a given result. Taking a root undoes the effect of exponentiation: if a base raised to an exponent equals a certain value, then the corresponding root of that value recovers the original base.

Structure of the Radical Symbol

Radical Symbol Structure

The radical symbol consists of a checkmark-like sign extending over a horizontal bar, with the quantity whose root is being taken placed beneath the bar and a small number, when present, nested in the notch of the checkmark to indicate which root is intended.

Radicand Identification

The radicand is the quantity that sits beneath the horizontal bar of the radical symbol, representing the value whose root is being sought. The entire expression under the bar is treated as a single quantity, and any operations within it must be completed before the root is applied.

Root Index Identification

The index is the small number nested in the notch of the radical symbol, indicating which root is being taken. An index of 2 indicates a square root, an index of 3 indicates a cube root, and any other whole number index indicates that corresponding root.

Implied Square Root Index

When no index is written on the radical symbol, an index of 2 is implied, meaning the expression represents a square root by default. This is the only index that may be omitted; every other root must display its index explicitly.

Reading Roots Aloud

Square Root Reading

A radical with an index of 2, or no index shown, is read as "the square root of" the radicand, reflecting its role as the inverse of squaring.

Cube Root Reading

A radical with an index of 3 is read as "the cube root of" the radicand, reflecting its role as the inverse of cubing.

Higher Root Reading

A radical with an index greater than 3 is read as "the nth root of" the radicand, using the ordinal form of the index number, such as "the fourth root of" or "the fifth root of," reflecting its role as the inverse of raising to that corresponding power.

Exact and Approximate Root Values

Exact Root Value

A root has an exact value when the radicand is a perfect power matching the index, so that a whole number or simple fraction, when raised to that index, produces the radicand precisely, with no rounding required.

Approximate Root Value

A root has only an approximate value when the radicand is not a perfect power matching the index, producing a non-terminating, non-repeating decimal that must be truncated or rounded to a chosen number of decimal places for practical use.