Table of Contents. Number and Operation. Geometry. Measurement. Lesson 1 Googols and Googolplexes Lesson 2 Space Voyager...

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2 Table of Contents Number and Operation Lesson Googols and Googolplexes OPERatIOns with ExPOnents Lesson 2 Space Voyager ScIEntIFIc notation Lesson 3 Taxing Experiences SaLEs Tax and Discounts Lesson 4 how the Earth Breathes PERcent IncREase and DecREase Lesson 5 Protecting Money Simple IntEREst Geometry Lesson 6 Digging Up the Roman Army PythaGOREan ThEOREM Lesson 7 The Golden Ratio TrigonOMEtRIc RatIOs Lesson 8 Red Eye DILatIOns Lesson 9 america s Main Street TransVERsals Lesson 0 Punxsutawney Phil REFLEctIOns Lesson around We Go RotatIOns Measurement Lesson 2 Revolution in Recording SuRFace area OF a CyLInDER Lesson 3 candle Making VOLuME OF cones and CyLInDERs Lesson 4 Scavenger Fish ChanGIng area and VOLuME

3 Algebra Lesson 5 Flying Down the Slopes ConcEPt OF slope Lesson 6 Touching the Sky Slope-IntERcEPt FORM Lesson 7 sidewalk Surfing StandaRD FORM Lesson 8 Moving Up Graph InequaLItIEs Lesson 9 the Great American Race EquatIOns with Two VaRIaBLEs Data Analysis and Probability Lesson 20 honest Abe ScattERPLOts Lesson 2 Zion DEPEnDEnt EVEnts Lesson 22 The Curse of the Bambino Math Tools PERMutatIOns Glossary Order of Operations Formulas and Trigonometric Table Perimeter, Area, and Volume Angle Pairs Fractions, Decimals, and Percents

Operations with exponents Googols and Googolplexes We are familiar with names for large numbers. We know that one million is a followed by six zeros,000,000.

But what would you call a followed by one hundred zeros?

When Kasner asked his nephew what the name should be, the young boy replied that it should be called a googol. As a number, a googol is written as 0 00.

4 Operations with exponents Googols and Googolplexes We are familiar with names for large numbers. We know that one million is a followed by six zeros,000,000. A billion has nine zeros, a trillion has twelve zeros, and a quadrillion has fifteen zeros. Next comes a quintillion, with eighteen zeros. But what would you call a followed by one hundred zeros? The name for this number was actually invented by the nine-year-old nephew of Edward Kasner, an American mathematician. When Kasner asked his nephew what the name should be, the young boy replied that it should be called a googol. As a number, a googol is written as Even greater than a googol is a googolplex, which is the number followed by a googol zeros! Get Started Use repeated multiplication to rewrite each expression in the first column of the chart below. Then multiply to find the value. The first one has been done for you Lesson : Operations with Exponents 5

5 Multiplying Powers with a Common Base Suppose you want to find the product of 3 2 and 3 4. The expressions 3 2 and 3 4 have the same base, or a common base. You can rewrite 3 2 as 3 3. And you can rewrite 3 4 as Two factors of three (3 2 ) multiplied by four factors of three (3 4 ) give you a total of six factors of three (3 6 ). The total number of factors equals the sum of the exponents: (2 4) 3 6 Suppose you want to find the product of 4 3 and 4 5. You can rewrite 4 3 as And you can rewrite 4 5 as Three factors of four (4 3 ) multiplied by five factors of four (4 5 ) gives us a total of eight factors of four (4 8 ). The total number of factors equals the sum of the exponents. So when you multiply powers having a common base (a), you add the exponents (m + n) to find the product: a m a n (m n) a. What is the product of 0 3 and 0 4? 6 Level H

6 Practice Use the formula a m a n a m n to find each product below m 4 m 5 Solve a Problem. When a number has no exponent written next to it, it is understood that the exponent is : 2 2 Use this fact to help you find the product below. It s a Fact! 0 24 septillion 0 27 octillion 0 30 nonillion 0 33 decillion 0 36 undecillion 0 39 duodecillion 0 42 tredecillion ? Lesson : Operations with Exponents 7

7 Dividing Powers with a Common Base Suppose you want to find the quotient The expressions 5 6 and 5 2 have the same base, or a common base. You can rewrite as Since is the same as, we can rewrite the expression above as (5 5) (5 5) This is equal to , or 54. So When we divide powers with a common base (a), we subtract the exponents (m n) to find the quotient: a m a n (m n) a It is also possible for a number to have a negative exponent. If you use the formula above to find the quotient 02 04, you can see that (2 4), or 0 2. So We can rewrite this expression as (0 0) (0 0) You can see that So to rewrite a power with a negative exponent, you can use this formula: a m a m 8 Level H

8 Show What You Know Use the formula a m a n a m n to write each quotient below Use the formula a m m a to rewrite each expression below Ben says that any number with an exponent of 0 is equal to. Is Ben correct? Explain why or why not. On Your Own Write down the three formulas you learned in this lesson. Explain in your own words why each formula works. Lesson : Operations with Exponents 9

9 Test Yourself. What is the value of 2 6? Find the value of the expression below. Show how you found your answer Which shows the product of 3 5 and 3 5? Which shows the quotient ? In your own words, explain why 6 2 = Which shows the product b 2 b b 4? b 6 7. Think Back In the expression 5 3, what is the base? What is the exponent? b 7 b 8 b 9 0 Level H

Black 4 Step Problem Solving. Astronomy and Large Numbers

Black 4 Step Problem Solving. Astronomy and Large Numbers Black 4 Step Problem Solving Astronomy and Large Numbers People have always been fascinated by the moon, the planets, and the stars. Hundreds of stories have been written to try and explain these lights