Mathematics for Technology II (Math 2131)

A radical is said to be in simplest form when:

1. The radicand has been reduced as much as possible.
2. There are no radicals in the denominator and no fractional radicands.
3. The index has been made as small as possible.

This section will cover part (3). For information on part (1) and part (3), follow the links.

Your last stop to simplifying radicals is knowing how to reduce the index, if applicable. To reduce the index, you’re probably better-off working with fractional exponents instead, since it makes more sense to apply the rules of exponents when reducing. Take, for example, the expression:

According to the power raised to a power exponent law, you multiply the exponents: 1/6 × 3 = 1/2.

Notice that the index has been reduced from 6 to 2 (only indices greater than 2 are displayed). This example is further explained in the video below along with another:

Try this out:   Reduce the expression

$\sqrt[6]{\frac{4x^6}{9}}$

Solution

$\frac{{\left[\left(2^2\right)·x^6\right]}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}}{{\left(3^2\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}}=\frac{{\displaystyle {\left(2^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}{\displaystyle ·}{\displaystyle {{\displaystyle \left(x^6\right)}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}}{3^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}=\frac{2^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}{3^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}·x=\boxed{\sqrt[3]{\frac{{\displaystyle 2}}{{\displaystyle 3}}}·x}$

At this point, you can rationalize the expression that’s boxed. If you do, your answer will look like this:

$\frac{\sqrt[3]{2}}{\sqrt[3]{3}}·x·\frac{\sqrt[3]{9}}{\sqrt[3]{9}}=\frac{\sqrt[3]{18}}{\sqrt[3]{27}}·x=\boxed{\frac{\sqrt[3]{18}·x}{3}}$

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