Partner Study Exciting Exponents! Follow the directions On this Power-point Complete Assignments Check with answer keys. Take Quizzes if prompted.

Okay everyone…. Mrs. Rudder has informed me that you are all independent learners and can work together to teach yourselves a lot of things. I am going to put you to the test….. The POWERFUL test… Yes, you guessed it, The powerful test of learning about exponents. Truthfully, they aren’t that bad. They are just a part of a systematic routine that is orderly and organized and has laws that must be followed so we are accurate and can explain things. Are you up for the challenge? (Actually, you better be, since you really have no choice in the matter! )

First of all…… You probably already know this…………….. Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 53. The "exponent", being 3 in this example, stands for however many times the value is being multiplied. The thing that's being multiplied, being 5 in this example, is called the "base". This process of using exponents is called "raising to a power", where the exponent is the "power". The expression "53" is pronounced as "five, raised to the third power" or "five to the third". There are two specially-named powers: "to the second power" is generally pronounced as "squared", and "to the third power" is generally pronounced as "cubed". So "53" is commonly pronounced as "five cubed".

Simplify (x3)(x4) (x3)(x4) = (xxx)(xxxx) = xxxxxxx = x7 When we deal with numbers, we usually just simplify; we'd rather deal with "27" than with "33". But with variables, we need the exponents, because we'd rather deal with "x6" than with "xxxxxx". Exponents have a few rules that we can use for simplifying expressions. Simplify (x3)(x4)    To simplify this, I can think in terms of what those exponents mean. "To the third" means "multiplying three copies" and "to the fourth" means "multiplying four copies". Using this fact, I can "expand" the two factors, and then work backwards to the simplified form: (x3)(x4) = (xxx)(xxxx)           = xxxxxxx           = x7 Note that x7 also equals x(3+4). This demonstrates the first basic exponent rule: Whenever you multiply two terms with the same base, you can add the exponents: ( x m ) ( x n ) = x( m + n ) Hey, I am being raised to the power of myself. So, if I am worth 6, then this would represent 6⁶.

Uh…. Not really. This seems kinda simple. However, we can NOT simplify (x4)(y3), WHY?....... because the bases are different: (x4)(y3) = xxxxyyy = (x4)(y3). Nothing combines! Pretty brainy, huh? Uh…. Not really. This seems kinda simple.

Here is an example of what I mean… Now…. We all know that exponents are pretty powerful but there is something else that is pretty powerful. That is the great PARENTHESIS!! ( ) Here is an example of what I mean… If you have 6a², it is simply 6 x a² and that is 6a² or 6 x a x a…… YOU ONLY SQUARE THE a! But what if….. YOU HAD (6a)²? (Remember parenthesis are like houses or “sets”. You have to keep the set together! ) That means you multiply the 6a by itself TWO times ….. That is a bigger value: (6a)(6a) = 36a²

9⁰=1 129⁰=1 Even 2,234,599⁰ = 1 Oh, One more thing….. You better believe in the power of ZERO. If a base is raised to the zero power it IS worth something. Only ONE thing. That’s right. ONE! 9⁰=1 129⁰=1 Even 2,234,599⁰ = 1

Evaluate or simplify the following: 16² 11³ 5⁴ + 3³ b⁰ 25c⁰ - 4 ¹ Exponent Expert Assignment One Let’s see how your expertise is right now….. Copy and solve the following problems in your spirals. Write the date and title your page, EXPONENT EXPERT ASSIGNMENT ONE. Work with your partner. Agree on your work and help each other. Get these checked before going to the next slide. Evaluate or simplify the following: 16² 11³ 5⁴ + 3³ b⁰ 25c⁰ - 4 ¹ Write 62 to the power of one. Evaluate its value Re-write using exponents: (-8)(-8)(-8)(-8) Re-write using exponents: (-6)(-6)(-6) (7)(7) (a)(z) (5a)³ vs. 5a³ (3y)² + x³ + 4y⁴ 24y³ + 3y³ + (4y)³ 6z⁶ - (2z)⁶

Talk over what you got incorrect and see if you can understand why! Get the answer sheet And Check your first set Of Twelve problems. Talk over what you got incorrect and see if you can understand why!

Laws of Exponents – A little Review The simplest law is the following: The exponent tells how many times to multiply the base number by itself. (We all probably know that!) Law #2: Any number to the power of 1 is itself. Law #3: Any number to the power of 0 is 1. Law #4: The next law pertains to the example below. We went over this but let’s review again…..When your bases are the same, you just add the exponents to simplify. LAW: SAME BASE/ADD EXPONENTS Example: x2x3 = (xx)(xxx) = xxxxx = x5 More examples: y³y⁴ = y⁷ BUT….be CAREFUL….If this is y³ + y⁴ you CANNOT add their exponents. Why? That little addition operation gets in the way! Let’s give the variable a value…. Let’s say y = 3 y³y⁴ = (3)(3)(3) TIMES(3)(3)(3)(3) = 3⁷ = 2187 BUT…. y³ + y⁴ = (3)(3)(3) + (3)(3)(3)(3) = (27) + (81) = 108

Complete your second set of problems on the next slide… Now…. Complete your second set of problems on the next slide… I hope I’m up for this! Sure…. Why not……

CHECK THIS SET BEFORE GOING TO SET TWO! Talk over things you missed. Exponent Expert Assignment Two In your spirals, go to a new page. Write the date and title this Exponent Expert Problems Assignment Two. Copy the problems and solve or simplify as much as you can. Check them when finished. SET ONE: 4⁰ + 7¹ 4x⁰y² + (c)(c) 1 ÷ 4³ + (2z)² (3/4) (x³x⁴) y⁷y³ + y³ (15a³)(-3a) (4c⁴)(ac³)(3a⁵c) a⁶b³ (a²b²) (-2.4n⁴)(2n¹)(2n)⁰ (x⁵y²)(-x⁶y) CHECK THIS SET BEFORE GOING TO SET TWO! Talk over things you missed. With the following problems, you should discover the 5th law of Exponents. Hint: Show the problems in expanded form to help you solve. Then write the law. See if you are correct! Check your answers when you and your partner agree. SET TWO: (6y²)² (2y⁴)³ (x²) – (x³)² (x²)⁵ (x³)²

Do you feel like experts yet? Video Time! https://rudderresponse.pbworks.com/w/page/86159641/Exponents Let’s watch a video here. This “funny” math guy will try to explain the law(s) you have been working with and have just discovered! Click on Mrs. Rudder’s link above and follow directions for this Assignment.

Law #5 When an exponent is raised to another exponent Or you could say…. When an exponent is raised to another power without a base inbetween, You simply multiply the exponents. BUT……remember there should only be one base! (xm)n = xmn (x2)3 = x2×3 = x6 In your spirals, write 5 examples of your own with numbers. Prove them using expanded form like this: (X³)² = X³⁽²⁾ = X⁶ X*X*X TWO TIMES = (X*X*X)(*X*X*X)= X⁶ Get your five examples checked. Label them “Proof of Law #5)

You can’t mess around with There is NO tolerance for that! Ready for another Law? You can’t mess around with EXPONENT And be lawless! There is NO tolerance for that!

Go to the candy bowl and you will see some cookies Go to the candy bowl and you will see some cookies. Help yourself to a couple. For real….

Law #6 Negative Exponents If your base is raised to a negative power you simply put that base and power in the denominator of a fraction with “1” in the numerator….. Example: 6⁻² = 1/6² or 1/36 x-n = 1/xn x-3 = 1/x3

Get it?

These worksheets are in a packet located in your green folders! Go to this “Math is Fun” Site and Practice. Practice together with your partner on this computer. See what kind of experts you are! http://www.mathopolis.com/questions/q.php?id=751&site=1&ref=/algebra/negative-exponents.html&qs=751_2024_2025_2026_752_2027_3147_3148_3149_3150 Hope you did well. Get help if needed. Now, Get the Negative Exponent Problem Solving Sheet. Complete that sheet and get it checked. Finally, get the “Laws of Exponents” Practice Page. Complete that page and get it checked. These worksheets are in a packet located in your green folders!

Here are the laws in review. Remember them… Now…. You should be an exponent EXPERT! Just like this dude here…but I think he hasn’t studied as much as you! Here are the laws in review. Remember them… HINT…HINT… Law Example x1 = x 61 = 6 x0 = 1 70 = 1 x-1 = 1/x 4-1 = 1/4 xmxn = xm+n x2x3 = x2+3 = x5 xm/xn = xm-n x6/x2 = x6-2 = x4 (xm)n = xmn (x2)3 = x2×3 = x6 (xy)n = xnyn (xy)3 = x3y3 (x/y)n = xn/yn (x/y)2 = x2 / y2 x-n = 1/xn x-3 = 1/x3