Simplest Form of a Radical

Simplification of radicals

The form of a radical expression may be changed without altering its value: \dfrac{1}{\sqrt2} can be changed to \dfrac{\sqrt2}{2}, for each equals .707\ldots

Study these changes of form:

1. \sqrt{36} = \sqrt{4 \cdot 9} = \sqrt4 \cdot \sqrt9 = 2 \cdot 3 = 6.
2. \sqrt8 = \sqrt{4 \cdot 2} = \sqrt4 \cdot \sqrt2 = 2\sqrt2.
3. More generally, \sqrt{a^2b} = \sqrt{a^2}\sqrt b = a\sqrt b.
4. \sqrt[3]{24} = \sqrt[3]{8 \cdot 3} = \sqrt[3]{8}\,\sqrt[3]{3} = 2\sqrt[3]{3}, and in general \sqrt[n]{a^nb} = a\sqrt[n]{b}.

A radical is in its simplest form when the radicand

• (a) is integral (no fraction under the radical sign),
• (b) contains no factor raised to a power equal to, or greater than, the order of the radical, and
• (c) is not raised to a power unless the exponent of the power and the index of the root are prime to each other.

The answer checker holds you to these rules: \sqrt{12} is marked "equal, but not in simplest form", and so is \dfrac{1}{\sqrt2}, because it leaves a radical in the denominator. Write 2\sqrt3 and \dfrac{\sqrt2}{2}. It also asks you to put a product of radicals under one sign: \sqrt{2}\cdot\sqrt{x} is written \sqrt{2x}.

Of (a), an integral radicand. Multiply both terms of the fraction under the sign so that the denominator becomes a perfect power: \sqrt{\tfrac23} = \sqrt{\tfrac69} = \tfrac13\sqrt6, \qquad \sqrt[3]{\tfrac{2}{9}} = \sqrt[3]{\tfrac{6}{27}} = \tfrac13\sqrt[3]{6}, \qquad \sqrt{\frac{a}{2b}} = \sqrt{\frac{2ab}{4b^2}} = \frac{1}{2b}\sqrt{2ab}.

Of (b), no factor to too high a power. Take out every factor whose power can be divided by the index: \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt3, \qquad \sqrt[3]{16a^4} = \sqrt[3]{8a^3 \cdot 2a} = 2a\sqrt[3]{2a}.

Of (c), the index and the power prime to each other. Divide the index and the exponent by a common factor: \sqrt[4]{9} = \sqrt[4]{3^2} = 3^{2/4} = 3^{1/2} = \sqrt3, \qquad \sqrt[6]{27x^3} = \sqrt[6]{(3x)^3} = \sqrt{3x}.

An expression under a radical must be simplified as a whole first: \sqrt{1 + \left(\tfrac12\right)^2} = \sqrt{\tfrac54} = \tfrac12\sqrt5, and \sqrt{a^2 + \left(\tfrac a3\right)^2} = \sqrt{\tfrac{10a^2}{9}} = \tfrac a3\sqrt{10}.

How to type radicals

• Square root: sqrt(12) or √12; a coefficient goes in front: 2sqrt(3) or 2√3.
• Cube root: cbrt(5) or ∛5. Any other root: root(x, 4) for \sqrt[4]{x}, root(2, 6) for \sqrt[6]{2}.
• Put the whole radicand in parentheses when it is more than one number or letter: sqrt(2x), cbrt(a + b), sqrt(2 - sqrt(2)).
• Fractional exponents use ^ with parentheses: x^(2/3).
• Letters under a radical sign stand for positive numbers, so \sqrt{x^2} = x.