Radicals and Fractional Exponents

Radicals

All the numbers of algebra are either real or imaginary. Real numbers are rational (an integer, or the quotient of two integers, like 7, \tfrac38, 4.237) or irrational. A rational number can be obtained from 1 by addition, subtraction, multiplication and division, which are therefore called the rational operations.

\sqrt2 = 1.414213\ldots is irrational: the decimal never ends and never repeats, so it cannot be the quotient of two integers. Even so, it can be shown exactly as a length: if the legs of a right triangle are each 1 inch, the hypotenuse is \sqrt2 inches.

Even roots of negative numbers, like \sqrt{-4}, are imaginary numbers; they will be treated later.

• A radical is an indicated root of any algebraic expression: \sqrt4, \sqrt3, \sqrt[3]{a}, \sqrt{x^2 - 5x + 6}.
• A surd is an irrational root of a rational number, like \sqrt3 or \sqrt[3]{7}. Not every irrational number is a surd: \pi = 3.14159\ldots, the ratio of the circumference of a circle to its diameter, is not.
• The index shows the order of the radical: in 5\sqrt[3]{7}, 3 is the index, the radical is of the third order, and 5 is the coefficient. The radicand is the expression under the radical sign.
• The principal root of a number is its one real root, if it has but one, or its positive real root, if it has two. The principal root of \sqrt[3]{-27} is -3; that of \sqrt[4]{16} is +2, not -2.

Fractional exponents

We assume the laws of integral exponents hold for fractional exponents too. Since x^{1/2} \cdot x^{1/2} = x^1 = x, the number x^{1/2} is another way of writing \sqrt x. In the same way x^{1/3} = \sqrt[3]{x}, and in general x^{1/n} = \sqrt[n]{x}. So 4^{1/2} = 2, 8^{1/3} = 2, (25a^2)^{1/2} = 5a.

Further, x^{2/3} = \left(x^{1/3}\right)^2 = \left(\sqrt[3]{x}\right)^2, and also x^{2/3} = (x^2)^{1/3} = \sqrt[3]{x^2}. For example 8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4, and also \sqrt[3]{64} = 4. In general x^{a/n} = \sqrt[n]{x^a} = \left(\sqrt[n]{x}\right)^a: the denominator of a fractional exponent is the index of the root, and the numerator is the power to which the radicand is raised. A negative exponent still means the reciprocal: 9^{-1/2} = \dfrac{1}{9^{1/2} = \dfrac13.

The real cube root of a negative number is negative: (-27)^{1/3} = -3, so (-27)^{2/3} = 9.

Examples. 16^{3/4} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8. \quad a^{3/5} = \sqrt[5]{a^3}. \quad 3m^{1/2}(n + 1)^{1/3} = 3\sqrt m\,\sqrt[3]{n + 1}.

Typing answers: when asked to write with radical signs, use sqrt(...), cbrt(...) and root(..., n): \sqrt[4]{a^3} is root(a^3, 4), and \sqrt[a]{t} is root(t, a). Numerical values are plain numbers like 27 or 1/4.