BM4.2 – Working with Radicals | For the Love of Maths

Chapter 4, Lesson 2

In this lesson students will:

understand how to interpret the relationship between the base and the power of an exponential expression
utilize the power laws and the BAII Plus calculator to simplify or evaluate exponential expressions

Working with Radicals

Interpretation of Fractional Powers

There may be instances in which we will need to work with expressions involving square roots or nth roots. Unfortunately, our calculator is not set up nicely to evaluate higher order radicals, so we will need to interpret them using fractional powers instead. Before reviewing this, let’s develop some terminology for radical expressions that will be helpful in the next few paragraphs. A radical expression is of the form $\sqrt[m]{a}$ , where $a$ can be thought of as the base expression, and $m$ is called the index. If $m$ is not written, it is understood to be the square root expression. For example, $\sqrt[3]{4}$ is the cube root of four, where $4$ is the base and $3$ is the index of the expression. In general, we cannot accept negative values under a root with even index; we can, however, accept negative values under a root with odd index. Your calculator will return Error 2 if it cannot evaluate an expression (try evaluating $\sqrt{-4}$ , for instance).

So how do we ask our calculator to evaluate $\sqrt[3]{4}$ if we do not have a cube root button? The key is to use the connection between radicals and fractional powers. Recall that in general we have

$\begin{aligned} a^{1/n} &= \sqrt[n]{a} \end{aligned}$

so this means we can evaluate this expression by re-writing it as $\sqrt[3]{4} = 4^{1/3} \approx 1.5874$ . Be mindful that you must use brackets around the fractional power for your calculator to evaluate correctly. The exact series of buttons to press is $4$ $( \; 1 \; \div \; 3 \; ) \; =$ .

Evaluate the following expressions using your calculator, and watch the video below to check your answers.

$\sqrt[5]{637}$
$-\sqrt{361}$
$\sqrt[4]{-25}$
$(\sqrt[3]{530})^{2}$

VIDEO

Properties of Radical Expressions

Provided you are willing to convert radical expressions to fraction powers, the five major power laws discussed in Lesson 4.1 can be use for simplification purposes. In addition to those laws, there are two other helpful properties that can be used (these are direct consequences of laws four and five):

A radical can be distributed to all terms connected by multiplication under the radical: $\sqrt[m]{a \times b} = \sqrt[m]{a} \times \sqrt[m]{b}$ .
A radical can be distributed to the numerator and denominator of a fraction: $\displaystyle \sqrt[m]{\frac{a}{b}} = \frac{\sqrt[m]{a}}{\sqrt[m]{b}}$ .

Let’s work through a few examples to make sure we are more comfortable with these kinds of expressions. The solutions will be given in the video below.

Evaluate: $\sqrt[3]{\sqrt{3}}$ .
Evaluate: $\displaystyle \sqrt[5]{\frac{16807}{32}}$ .
Simplify: $\sqrt{4x+4}$ .
Simplify: $\displaystyle \frac{(2ab)^{3}a^{-1}\sqrt{b}}{\sqrt[3]{ab} \; a^{-2}}$ .