More Simplification; Radicals Under One Sign

Radicands that contain radicals

The rules of simplest form apply to a radicand like 8 + 4\sqrt5 too: take out any factor of the whole radicand that is a perfect power of the order of the radical. \sqrt{12 + 4\sqrt5} = \sqrt{4\left(3 + \sqrt5\right)} = 2\sqrt{3 + \sqrt5}, \qquad \sqrt[3]{24 - 8\sqrt3} = \sqrt[3]{8\left(3 - \sqrt3\right)} = 2\sqrt[3]{3 - \sqrt3}. A letter that is a factor of every term comes out the same way: \sqrt{a^2 + a^2\sqrt2} = a\sqrt{1 + \sqrt2}, and a fraction is cleared as before: \sqrt{\dfrac{a^2 + a^2\sqrt5}{2}} = \dfrac a2\sqrt{2 + 2\sqrt5}.

Reversing the process

To put an expression entirely under the radical sign, raise the coefficient to the power shown by the index and multiply it into the radicand: 3\sqrt2 = \sqrt9\,\sqrt2 = \sqrt{18}, \qquad 2\sqrt[3]{3} = \sqrt[3]{8}\,\sqrt[3]{3} = \sqrt[3]{24}, \qquad a\sqrt{\frac{b}{a}} = \sqrt{\frac{a^2b}{a}} = \sqrt{ab}. Here the radicand need not be in simplest form; the whole answer is a single radical with nothing outside it.

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.

In this lesson some exercises print a radicand that is negative (such as 25\sqrt5 - 100); its square root would be imaginary, so the terms have been taken in the opposite order.