# Mathematics I (Math 1132) Study Force Academy
Durham College, Mathematics
Free • 42 lessons
• 0 quizzes
• 10 week duration
• ##### Measurements

An introduction to numerical computation. Emphasis is placed on scientific and engineering notation, the rule of significant figures, and converting between SI and Imperial units.

• ##### Fractions, Percentage, Ratios and Proportion

Emphasis here is placed on understanding fractions, percent, and using ratios to compare quantities and set up proportions to solve problems.

• ##### Geometry

This unit focuses on analyzing and understand the characteristics of various shapes, both 2D and 3D.

## Mathematics I (Math 1132)

### Laws of Exponents

Before we can do anything further with algebraic expressions, including multiplying or dividing monomials and polynomials, it’s critical that you know the laws of exponents. Of course, you could probably get away without knowing these laws formally, but then you wouldn’t have a strong foundation. A big part of your studies is being able to communicate your findings, both on paper and in speech. Thus, knowing the terminology will help tremendously.

We’ll use the laws of exponents mainly to simplify expressions, to make them easier to work with in later computations, such as solving equations containing exponents. The easiest of these laws to grasp are the zero exponent rule and negative exponent rule. The video below will explain what to do when you encounter zero and negative exponents.

The other exponent laws you will encounter are summarized underneath:

# 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:

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

Answer:

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

(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:

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

Answer:

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

(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:

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

Answer:

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

(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 more thorough explanation of these laws 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.