# 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)

### Review of the Exponent Laws

Radicals (√, ∛, ∜, etc.) are an extension of the exponents laws you learned in Part 1 of this course. This section is solely dedicated to the exponent laws. The connection between radicals and exponents is made in the next section, though it’s highly advised that you review these first as they’re easily forgettable! A summary of the rules are outlined below with examples.

# Product rule:

Rule:

${x}^{a}·{x}^{b}={x}^{a+b}\phantom{\rule{0ex}{0ex}}$

Examples:

${y}^{2}{y}^{4}={\mathbit{y}}^{\mathbf{6}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(b\right){\left(b\right)}^{3}{\left(b\right)}^{4}={\mathbit{b}}^{\mathbf{8}}\phantom{\rule{0ex}{0ex}}$

You try:

${m}^{3}{m}^{n}\phantom{\rule{0ex}{0ex}}$

${m}^{3+n}\phantom{\rule{0ex}{0ex}}$

(notice how the exponent 3 and n are not like terms so we leave it as 3 + n.

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# Quotient rule:

Rule:

$\frac{{x}^{a}}{{x}^{b}}={x}^{a–b}\phantom{\rule{0ex}{0ex}}$

Examples:

$\frac{{y}^{5}}{{y}^{2}}={y}^{3}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{{x}^{2}{y}^{5}}{x{y}^{3}}=x{y}^{2}\phantom{\rule{0ex}{0ex}}$

You try:

$\frac{{a}^{5n}}{{a}^{2n}}\phantom{\rule{0ex}{0ex}}$

${a}^{3n}\phantom{\rule{0ex}{0ex}}$

(notice how the exponents 5n and 2n are like terms, so we subtract the coefficients only).

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# Power of a power rule:

Rule:

${\left({x}^{a}\right)}^{b}={x}^{a×b}\phantom{\rule{0ex}{0ex}}$

Examples:

${\left({x}^{2}\right)}^{5}={x}^{10}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\left(xy\right)}^{4}={x}^{4}{y}^{4}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

Test:

${\left(3y\right)}^{3}\phantom{\rule{0ex}{0ex}}$

$27{y}^{3}\phantom{\rule{0ex}{0ex}}$

(Students will commonly mistaken the power with multiplication, for example, multiply the exponent 3 by the base 3 instead of 3 × 3 × 3 = 27)

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A thorough explanation of these laws and more are provided in the video:

Common error:

${\left(x+y\right)}^{n}\ne {x}^{n}+{y}^{n}\phantom{\rule{0ex}{0ex}}$

Students of all math backgrounds make this common mistake. Remember, you can only distribute the exponent n if what’s inside the brackets is a monomial; x and y are two separate terms, hence a binomial. You could distribute the n in the following cases:

${\left(xy\right)}^{n}\to {x}^{n}{y}^{n}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\left(x\left(x+y\right)\right)}^{n}\to {x}^{n}{\left(x+y\right)}^{n}\phantom{\rule{0ex}{0ex}}$

In the lesson to come, you will learn how to handle expressions like (x + y)² via a technique called expanding. For a list of other common math errors, watch this link.

Now it’s time to put your knowledge of exponents to the test. The video below shows three complicated examples that require you use several of the exponent laws to simplify a single expression.