# Mathematics for Technology II (Math 2131)

Durham College, Mathematics
Free
• 36 lessons
• 0 quizzes
• 14 week duration
• ##### Solving Systems of Equations

This unit introduces how to systematically solve a system of equations, namely linear equations. Examples of non-linear systems, including systems of 3 unknowns will be of emphasis.

• ##### Graphs of Trigonometric Functions

The unit focuses primarily on how to graph periodic sinusoidal functions, and how to identify features of a waveform to produce an equation by inspection.

• ##### Polar Coordinate Functions

An introduction to the polar coordinate system.

• ##### Complex Numbers

This unit is an extension of what was introduced in Math 1131. To learn how to work with radicals, knowing your exponent laws in crucial. Hence, this unit begins with a thorough review.

• ##### Logarithmic Functions

This chapter introduces you to exponential functions, and how they can be solved using logarithms.

• ##### Trigonometric Identities and Equations
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• ##### Analytic Geometry
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## Mathematics for Technology II (Math 2131)

### Simplifying Radicals – Rationalizing the Index

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.

$\therefore {x}^{1}{2}}=\sqrt{x}\phantom{\rule{0ex}{0ex}}$

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

$\sqrt[6]{\frac{4{x}^{6}}{9}}\phantom{\rule{0ex}{0ex}}$
Solution

$\frac{{\left[\left({2}^{2}\right)·{x}^{6}\right]}^{1}{6}}}{{\left({3}^{{2}}\right)}^{1}{6}}}=\frac{{\left({2}^{2}\right)}^{1}{6}}·{\left({x}^{6}\right)}^{1}{6}}}{{3}^{1}{3}}}=\frac{{2}^{1}{3}}}{{3}^{1}{3}}}·x=\overline{)\sqrt[3]{\frac{2}{3}}·x}\phantom{\rule{0ex}{0ex}}$

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

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

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