Hi Everyone!
On this page you will find some material about Lesson 1. Read through the material below, watch the videos, and follow up with your instructor if you have questions.
Lesson 1: Properties of Integer Exponents & Addition and Subtraction of Rational Expressions
Table of Contents
Resources
In this section you will find some important information about the specific resources related to this lesson:
- the learning outcomes,
- the section in the textbook,
- the WeBWorK homework sets,
- a link to the pdf of the lesson notes,
- a link to a video lesson.
Learning Outcomes.
- Simplify expressions with exponents.
- Apply the properties to rewrite the expression with positive exponents only.
- Simplify rational expressions.
- Add and subtract rational expressions with like and unlike denominators.
Topic. This lesson covers
Section 4.1: Properties of Integer Exponents, and
Section 5.3: Addition and Subtraction of Rational Expressions.
WeBWorK. There are four WeBWorK assignments on today’s material:
IntegerExponents
ReducingRationalExpressions
AddRationalExpressions
AddRationalExpressions2
Lesson Notes.
Video Lesson.
Video Lesson 1 – part 1 (based on Lesson 1 Notes – part 1)
Video Lesson 1 – part 2 (based on Lesson 1 Notes – part 2)
Video Lesson 1 – part 3 (based on Lesson 1 Notes – part 3)
Warmup Questions I
These are questions on fundamental concepts that you need to know before you can embark on this lesson. Don’t skip them! Take your time to do them, and check your answer by clicking on the “Show Answer” tab.
Warmup Question 1
Find $4\cdot 3$ and $4^3$.
Show Answer 1
$$4\cdot 3=\underbrace{4+ 4+ 4}_{3\text{ counts the number of 4’s being added}} = 12$$
$$4^3=\underbrace{4\cdot 4\cdot 4}_{3\text{ counts the number of 4’s being multiplied}} = 64$$
Quick Intro I
This is like a mini-lesson with an overview of the main objects of study. It will often contain a list of key words, definitions and properties – all that is new in this lesson. We will use this opportunity to make connections with other concepts. It can be also used as a review of the lesson.
A Quick Intro to Integer Exponents
Key Words. Exponentiation, base, exponent, power.
In the expression $b^n$, $b$ is called the base and $n$ is called the exponent or power. The operation is called exponentiation.
$\bigstar$ When $n$ is positive, the exponent $n$ is the number of $b$’s being multiplied.
$$b^n=\underbrace{b\cdot b \cdots b}_{n\text{ times}}$$
In the exponential expression $2^5$, the base is $2$ and the exponent is $5$.
$$2^5 =\underbrace{2\cdot 2 \cdot 2 \cdot 2 \cdot 2}_{5\text{ times}} = 32$$
What if the exponent $n$ is zero? Then $b^0=1$ for $b\neq 0$.
What if $n$ is negative? $b^{-n} = \dfrac{1}{b^n}$ for $b\neq 0$.
$\bigstar$ Properties:
- $b^mb^n = b^{m+n}$
- $\dfrac{b^m}{b^n} = b^{m-n}$
- $(b^m)^n = b^{mn}$
- $b^nc^n = (bc)^n$
- $\dfrac{b^n}{c^n}= \left(\dfrac{b}{c}\right)^n$
$\bigstar$All expressions can be written with positive exponents only.
Video Lesson I
Many times the mini-lesson will not be enough for you to start working on the problems. You need to see someone explaining the material to you. In the video you will find a variety of examples, solved step-by-step – starting from a simple one to a more complex one. Feel free to play them as many times as you need. Pause, rewind, replay, stop… follow your pace!
Video Lesson – part 1
A description of the video
In the video you will see how to simplify:
- $2^3 2^4$
- $(4^2)^3$
- $2^35^3$
- $\dfrac{5^3}{4^3}$
- $(3x^5y^3)^2(2x^6y)^3$
- $\dfrac{(3x^5y)^2}{(9xy^2)^2}$
- $\left(\dfrac{2}{3}\right)^{-3}$
- $\dfrac{x^{-3}y}{x^2y^{-2}}$
- $3\left(\dfrac{2x^{-2}y^3z^{-5}}{3xy^2z^{-2}}\right)^2\left(\dfrac{-x^{2}y^{-2}}{zy}\right)^{-3}$
Try Questions I
Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. We encourage you to try the Try Questions on your own. When you are done, click on the “Show answer” tab to see if you got the correct answer.
Try Question 1
Find $5^3$.
Show Answer 1
$$5^3 = 5\cdot 5 \cdot 5 = 125$$
Try Question 2
Simplify $$(-2x^5y^2)^3(3x^6y^5)^2.$$
Show Answer 2
\begin{align*}& (-2x^5y^2)^3(3x^6y^5)^2 \\ = & (-2)^3(x^5)^3(y^2)^3(3)^2(x^6)^2(y^5)^2\\ =&-8x^{15}y^69x^{12}y^{10} \\ = & (-8\cdot 9)(x^{15}x^{12})(y^6y^{10})\\ =& -72x^{27}y^{16}\end{align*}
Try Question 3
Use Property (5), $\dfrac{b^n}{c^n}= \left(\dfrac{b}{c}\right)^n$, to simplify
$$\dfrac{(3x^5y)^2}{(9xy^2)^2}.$$
Write your final answer with positive exponents only.
Show Answer 3
\begin{align*} & \dfrac{(3x^5y)^2}{(9xy^2)^2} \\= &\left(\dfrac{3x^5y}{9xy^2}\right)^2 \\ =& \left(\dfrac{x^4}{3y}\right)^2\\ =& \dfrac{(x^4)^2}{(3y)^2}\\=&\dfrac{x^8}{9y^2}\end{align*}
WeBWorK I
You should now be ready to start working on the WeBWorK problems. Doing the homework is an essential part of learning. It will help you practice the lesson and reinforce your knowledge.
WeBWorK I
It is time to do the homework on WeBWork:
IntegerExponents
When you are done, come back to this page for the Exit Questions.
Exit Questions I
After doing the WeBWorK problems, come back to this page. The Exit Questions include vocabulary checking and conceptual questions. Knowing the vocabulary accurately is important for us to communicate. You will also find one last problem. All these questions will give you an idea as to whether or not you have mastered the material. Remember: the “Show Answer” tab is there for you to check your work!
Exit Question 1
- How is the negative exponent related to reciprocals? Give an example.
- How are positive and negative exponents used in science to express large or small numbers?
- What is the purpose in writing numbers this way?
$\bigstar$ (1) Simplify $$\left(\dfrac{3a^{-3}}{b^{-5}}\right)^{-3}.$$ Write your final answer with positive exponents only.
$\bigstar$ (2) Simplify $$\dfrac{36x^5y^{10}}{70x^{15}y^5}.$$ Write your final answer with positive exponents only.
Show Answer 1
$\bigstar$ (1)
\begin{align*} & \left(\dfrac{3a^{-3}}{b^{-5}}\right)^{-3}\\= &\dfrac{(3a^{-3})^{-3}}{(b^{-5})^{-3}}\\=&\dfrac{3^{-3}(a^{-3})^{-3}}{b^{15}}\\=&\dfrac{a^9}{3^3b^{15}}\\=&\dfrac{a^9}{27b^{15}}\end{align*}
$\bigstar$ (2)
\begin{align*} & \dfrac{36x^5y^{10}}{70x^{15}y^5}\\=& \dfrac{2\cdot 18y^{10-5}}{2\cdot 35x^{15-5}}\\=&\dfrac{18y^5}{35x^{10}}\end{align*}
