# 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 Denominator

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 (2). For information on part (1) and part (3), follow the links.

You’ll rarely find written in math literature a radical placed in the denominator of a quotient. That’s because most mathematicians will rationalize the denominator as a final touch to before recording any equation. Take, for example, the expression:

$\frac{5}{\sqrt{x}}\phantom{\rule{0ex}{0ex}}$

While this may appear to be simplified, there’s still one more step which involves rationalizing the denominator. To do this, you want to multiply the both the numerator and denominator by a factor that rid the denominator of the radical. Said differently, you want to multiply and divide the expression by a factor that rid the denominator of the radical.  This is a relatively simple procedure if the radical is a 2nd root, but it becomes more difficult for larger indices. The complete procedure is shown below; notice that the factor we chose to multiply and divide the expression was √x and 1/√x, respectively.

$\frac{5}{\sqrt{x}}×\frac{\sqrt{x}}{\sqrt{x}}\to \frac{5\sqrt{x}}{\sqrt{x·x}}\to \frac{5\sqrt{x}}{\sqrt{{x}^{2}}}=\overline{)\frac{5\sqrt{x}}{x}}\phantom{\rule{0ex}{0ex}}$

For more examples, watch part 1 and 2 of the series linked below:

• Part 2 can be found here, and it’s recommended that you watch it because it covers binomial radicands. Rationalizing a denominator that possesses a binomial requires that you know a familiar technique picked up earlier in this course known as multiplying by the conjugate. Albeit, this only works when the index is 2.