Radicals of Different Orders; Products of Polynomials

Radicals of different orders

To multiply radicals of different orders, first change them to the same order with fractional exponents, using the least common multiple of the indices:

Example 1. \sqrt2 \cdot \sqrt[3]{5} = 2^{1/2} \cdot 5^{1/3} = 2^{3/6} \cdot 5^{2/6} = \sqrt[6]{2^3 \cdot 5^2} = \sqrt[6]{200}.

Example 2. \sqrt[3]{a^2} \cdot \sqrt a = a^{2/3} \cdot a^{1/2} = a^{7/6} = a\sqrt[6]{a}.

Polynomials containing radicals

Multiply term by term, as with any polynomials, simplifying each product and collecting similar radicals:

Example 3. \left(2\sqrt3 - \sqrt2\right)\left(\sqrt3 + 2\sqrt2\right) = 6 + 4\sqrt6 - \sqrt6 - 4 = 2 + 3\sqrt6.

Example 4. \left(\sqrt7 + \sqrt3\right)\left(\sqrt7 - \sqrt3\right) = 7 - 3 = 4. The product of a sum and a difference has no radical left in it, a fact used in the next lessons.

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.

Write every product of radicals as one radical: \sqrt2\,\sqrt[3]{5} should be given as \sqrt[6]{200}.