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Lesson 3: Multiplying Powers With the Same Base

Introduction

Today we will learn through indirect method, which means that I will ask a lot of questions and your answers will get you to the results. I need everyone to raise their hand before answering for this to work. To come up with some of the answers you may need to think or research on computer. This is part of life of a mathematician. Confusion is good, it precedes a realization.

First we will go through a concrete example from pure mathematics 5^2*5^3 and discover a formula to multiply exponents with the same base.

Then we will apply it to real life example from science which is bigger than life. You’ll find out how many grains for sand we have on this planet.

Online exercises will be done another time, although they are on this web site under lesson 3b. You can take a look at them if you want to prepare yourself for the future lessons.

Final activity is to do some work on the worksheet. When you have questions, you can ask me.

Today’s lesson follows the discovery method where there are lots of questions that I ask and you can ask me lots of questions in the process of your discovery.

You can send me comments on this web page.

Now, you can go to “Launch” to continue with the material, unless you want to see the lesson goals and main points.

The main points for what you are learning today:

• a^m * a^n = a^(m+n)  example: 5^2*5^3 = 5^2+3 = 5^5. If you multiply two exponents with the same base, you add the exponents together. If you need to remember one thing from this lesson. Remember this.
• Why does that work? Write it out and see. 5^2 * 5^3 = 5*5 * 5*5*5 this is total of 5 times. i.e. 5^5
• What about negative exponents? Formula works in the same way. 5^2 * 5^-3 = 5^-1

Some of the exercises will need you to use these skills:

• (5^5)5. These are exponents of exponential expressions to a power. The shortcut formula is: 5^(5*5)=5^25. The general form of this is (a^m)^n=a^(m*n).
• Remember to simplify the expressions and also remember that a^-2=1/a^2. Also the reverse applies: 1/a^-2*a^3=a^2*a^3=a^2+3=a^5
• for example to simplify ((a^-1)(a^3))5 by writing it out a^(-2*5) * a^(3*5) = a^-10 * a^15 = a^5

What does pure mathematics mean?

Now we will find out what is the general rule for multiplying exponents with same base. We’ll make sure you understand how it works and why.

1. Write out 3^4*3^2 = 3*3*3*…
2. How many times do you multiply 3 by itself?
3. Write it again as exponent of base 3. 3^? What is “?”
4. What did you find out? Do you notice that the new exponent is a sum of the two exponents we started with.
5. What would be a generic formula you can deduce from this?

What if you have a as base and variables m and n as exponents?

1. Write out a^m*a^n. (a*a*a*… m times) (a*a*a* n times)
2. Write as one exponent with base a. What is the exponent? Could it be sum of m and n?
3. What did you find out? This is the general formula: a^m * a^n = a^(m+n)

Watch the first 4 minutes of this video for the general formula explanation:

3. Do we have more grains of sand on earth than there are stars in the universe?

There are 10^20 stars in the universe.
A cubic meter of sand has 10^9 grains of sand.

If all sand in the world was combined to one huge beach, its dimensions would be 1m*100m*100,000km.

 

How would you find out whether there are more stars in the universe than sand grains in the world?

Work on this on your own for two minutes and then ask questions. To arrive at the answer through a logical set of steps, you need to ask these questions:

1. What are the relevant numbers and units we need in order to solve this problem?
2. In order to know the volume of sand in cubic meters, you will need to know
   a) How much is 100,000km in meters? 1km=1000m
   b) What are the three amounts in the picture in scientific notation
   c) How many cubic meters there is sand in the world. Calculate.
3. How do you get the total number of grains of sand in the world? Multiply by cubic meters of sand in the world times number of grains in a cubic meter of sand.
4. Is the end result larger or smaller than the number of stars in the universe?

Work on the worksheet and ask questions if you don’t know how to solve it.
Examples of multiplying exponents with the same base:

• 10^4 *10^7
• 3^4 + 3^3
• a^m * a^n
• x^y * x^z
• Scientific notation: (3 * 10^5) (5 * 10^-12)=(3 * 5) (10^5 * 10^-12)
• (1.13 * 10^-17)(9.98 * 10^5)(3.34 * 10^22)
• Convert to same base exponent multiplication:
  2^8 + 2^8 = 2(1^8+1^8) = 2(2^8) = 2^1*2^8 = 2^9
• Convert to same base exponent multiplication:
  2^9 / 2^10= 2^9 * 2^-10 = 2^-1 = 1/2

• a^m * a^n = a^(m+n)
• write an expression with same base exponents and you can use this rule
• Simplify algebraic expression as much as possible ax^n*bx^m = ab(x^n*x^m) = ab(x^(n+m)) and ax^n*by^m = ab(x^n*y^m)

• Complete the worksheet.

• We will review this lesson and prepare for the quiz.