Root (mathematics)

Root (mathematics), term used to indicate a number that when repeatedly multiplied by itself results in a second number. For example, 2 is a third root of 8 (written ) because 2 multiplied by itself three times yields 8 (2 × 2 × 2 = 8). Taking the root of a number is therefore the opposite of raising a number to a power: since 3 raised to the second power is equal to 9 (3² = 3 × 3 = 9), 3 is a second root of 9 ( = 3). Similarly, ⁴Â = 2 because 2 × 2 × 2 × 2 = 16 and = 10 because (10 × 10 × 10 = 1000).

The √ symbol is called a radical sign. The small raised number, or index, of the root is the number of times that the root appears in the multiplication. The second root, also known as the square root, is often written without an index: is the same as Ã. Third roots are often called cube roots. Roots can also be indicated using fractions as exponents: is identical to x^‚.

Negative numbers can be roots, and some numbers have two square roots, one positive and one negative. For example, the cube root of –8 is –2 because –2 × –2 × –2 = 8, and the numbers 2 and –2 are both square roots of 4 because 2 × 2 = 4 and –2 × –2 = 4.

In computation, roots obey certain rules called the laws of radicals:

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(1)

For example, .

(2)

For example, .

(3)

For example, .

(4)

For example, .

(5)

For example, .

The root of any number with an index of 0 is undefined. The root of any number with an index of 1 is equal to itself: .

A second mathematical use of the word “root” involves algebraic equations such as 2x = 4. When a quantity inserted in place of the unknown quantity (x) renders the equation a true statement, that quantity is called a root of the equation. For 2x = 4 the root would be 2.

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