Topic: Power of numbers where the exponent
is a whole number
Teacher: Ramsel Eclarin
10 June 2019
- Learning Standards / Indicators
Strand 1: Numbers and Operations
Standard M1.1: Understanding diverse methods of presenting numbers and their application in real life
For Grades 10-12:
- Show relationships of various numbers in the real number system.
- Have concepts of absolute values of real numbers.
- Have concepts of real numbers expressed in exponential notation with rational indices, and real numbers expressed in radicals.
Strand 1: Numbers and Operations
Standard M1.2: Understanding results of operations of numbers, relationships of operations, and application of operations for problem-solving
For Grades 10-12:
- Understand concepts and find results of addition, subtraction, multiplication and division of real numbers; understand real numbers expressed in exponential notation with rational indices, and real numbers expressed in radicals.
Strand 1: Numbers and Operations
Standard M1.3: Use of estimation in calculation and problem-solving
For Grades 10-12:
Find estimates of real numbers expressed in radicals and real numbers expressed in exponents through appropriate calculation methods.
- Key Concept
Exponents or indices are being called the powers of numbers. Say a number, n , to the power of 2 is also referred to as n squared. The number n raised to the power of 3 is called n cubed. n is called the base number. Calculating an exponent is simply multiplying the base number by itself.
In this lesson, be able to work with positive exponents and positive base numbers including negative base index. The exponent tells you how many times to multiply the number by itself. Example, three to the power of four, or 34, will be: 3 x 3 x 3 x 3 = 9 x 9 = 81
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- Learning Objectives: Students will be able to
- Understand very well about properties of exponents in the number system.
- Interpret what same base means, 3-1means, fractional exponent means.
- Solve problems as indicated herein.
- Apply the properties in real life and in written examinations.
- Learning Activities
Warm-Up
- Roll call
- Course Requirements
- Math resources
- Math tools
- Classroom Rules
- Q and A
Math Vocabulary listing
- Base
- Exponent
- Positive integers
- Negative Integers
- Index
- Indices
- Negative exponent
- Fractional exponent
- Zero power
- Non zero number
(List and organize these words in your lesson Notebook)
Presentation
PROPERTIES OF EXPONENTS
Take a look:
2
2 × 2 = 4
2 × 2 × 2 = 8
2 × 2 × 2 × 2 = 16
2 × 2 × 2 × 2 × 2 = 32
Writing out all these 2s gets boring quickly. Who wants to write out twenty 2s, all multiplied together? (If this is you, please put your hand down. No one can see you right now anyway.)
Thankfully, there’s a shortcut. We write 2n, pronounced “2 to the n,” “2 raised to the n,” or “2 to the nth power,” which all mean n copies of 2 multiplied together. And to help you remember that we’re “raising it,” we even literally raise it up a little bit next to the number we’re multiplying. Aren’t mathematicians thoughtful? They even sent you flowers on your birthday. Remember that?
If we’ve got 2n, that little n is called an exponent or power, 2 is called the base, and the process of raising a number to a power is called exponentiation. The numbers 2, 22, 23, and so on are called powers of 2. If you see something like 2love, that’s the power of love.
Be Careful: When raising a negative number to a power, keep careful track of your negative signs. Clip and tag them if you have to. If it’s the negative number that’s being raised to the power, we get one thing:
(-2)4 = (-2)(-2)(-2)(-2) = 16
If not, we group it differently and get something else:
-24 = -(24) = -16
Sample Problem
Pat wrote -32 = 9. What did Pat do wrong?
The negative sign isn’t being squared, so the answer should be -9. It would only be positive 9 if we had (-3)2. We’re really, really sorry if your name is Pat. It’s a total coincidence.
A Little Bit About Zero
If we raise 0 to any positive exponent, we still get 0. This makes sense, because if you multiply one or more copies of 0 together, you’ll just get 0. Turns out it’s hard for 0 to become anything other than 0. Even if he really applies himself.
Any nonzero number raised to the 0 power is 1. Think about it this way:
24 = 16
23 = 8
22 = 4
21 = 2
As the exponent drops by 1, the answer is divided in half. If we drop the exponent by 1 once more and divide the answer in half again, we get 20 = 1. We can’t believe how many times you just dropped that exponent. Can’t you be more careful?
20 = 1
30 = 1
150 = 1
(-36.25)0 = 1
It’s 1s all the way down: raise any number to the power of 0, and the answer is 1.
Well, except for one weird exception. What’s 00? Zero is a troublesome number. We want 0 raised to any power to be 0, but we also want any number raised to the 0 power to be 1. There’s no way to win! This means that 00 is undefined. If it’s not too late, don’t think about this too hard. It’ll make your head hurt.
Multiplication
What is 25 × 27?
This means that you need to multiply 5 copies of 2 together, and then multiply that result by 7 copies of 2. That’s a total of 12 copies of 2. So 25 × 27 = 212. Why so many copies of 2? What are you, passing them out at a meeting?
If we have the same base with two different exponents and we’re multiplying these numbers, as in the above example, the exponents get added together. In symbols, if a, b, and c are real numbers, then: ab × ac = a(b + c)
Negative Exponents
So far, we’ve only looked at exponents that are positive integers. Let’s try to figure out what a number would be when raised to a negative exponent.
Suppose we want to understand what 3-1 means. Let’s use what we know about multiplying exponents. Since we add exponents during multiplication, 31 × 3-1 would be 31 + (-1) = 30 = 1. This tells us that 3-1 is the multiplicative inverse, or reciprocal, of 3. So .
Now what happens if we take bigger powers? Like 5-7, for example. In this case, we’ll look at 57 × 5-7 = 57 + (-7) = 50 = 1. So 5-7 is the same as (1/5)7.
Division
What’s 25 ÷ 22?
This means , so we’re just canceling out two of our 2s.
After reducing, our fraction equals 23.
In general, ab ÷ ac = a(b – c), because we start out with b copies of a, divide out c copies, and are left with b – c copies.
Heads up, though: a can’t be 0.
Notice that if b > c, you’re left with a positive exponent. But if b < c, you have a negative exponent.
Exponentiation
What is (25)3?
This really means (2 × 2 × 2 × 2 × 2)3. You can’t just add the 5 and the 3 together in this instance, because what we’re actually being asked to do is take 3 copies of (2 × 2 × 2 × 2 × 2), or 15 copies of 2 multiplied together. (25)3 = 25 × 3 = 215
So, in general, (ab)c = ab × c.
Raising Products to a Power
What’s (6 × 7)3?
Obviously we could just multiply 6 by 7 to get (42)3, but let’s see what happens if we leave ’em separated.
(6 × 7)3 = (6 × 7)(6 × 7)(6 × 7) = 63 × 73.
In general, if a and b are real numbers and c is a whole number, (a × b)c = ac× bc.
Raising Quotients to a Power
If a and b are real numbers and c is a whole number, . Just slap that exponent on the numerator and the denominator separately.
Practice
Sample Problem 1
What’s 42 ÷ 44?
This translates to: 42-4
4-2
Sample Problem 2
What’s 63 ÷ 67?
What this really means is “3 copies of 6 divided by 7 copies of 6”:
Cancel out 3 copies of 6 from the top and bottom of the fraction to get :
Answer: ______________
Production
Get ready for : Quiz Number 1.
Wrap-Up
Be careful: In order to use the properties above, the base of the exponents has to be the same. For example, we can’t combine 43 and 52. That’s unfortunately as nice as it gets with exponent notation. Which isn’t very nice.
Learning Materials
Education Press write up for Summary in Math Lessons
Whiteboard presentation solving
SPM 5-5 Textbook
https://www.mathsisfun.com
Evaluation
Checking of Quiz No. 1 on 18 June 2019
(20 items). One point for every correct answer.
Homework: Power of numbers where the exponent is a fractional index
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