Grade 6 Math Circles. Exponents


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1 Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles November 4/5, 2014 Exponents Quick Warmup Evaluate the following: Multiplication Where does the idea of multiplications stem from? Multiplication is simply a shorter way of writing a repeated addition. For example = 3 4, here we have the number 3 added with itself 4 times. So we simplify this to 3 times 4. Looking at the questions in the quick warm up write the additions ad multiplications and the multiplications as repeated addition. Exponents An exponentiation is a repeated multiplication. Similar to how a multiplication is a repeated addition. Remember, 5 3 is simply Similarly an exponentiation, 5 3 is simply
2 As shown in the picture above, we call the lower number the base, the upper number the exponent and when refering to the base and exponent as a whole we will say the power. When we see this notation we say Two to the exponent three. Note: the second and third exponents are often referred to as squared and cubed, respectively. So we might say two cubed instead of Two to the exponent three. Examples: Write the following as a multiplication then evaluate = = Write the following numbers as exponents with the given bases 1. 32, base 2 = , base , base , base , base 4 Order of Operation: BEDMAS If you are given , do you do the + or the first? You do the first. We have these order of operations to make sure everyone calculates the same way. If there was no defined order then someone could do: = 7 2 = 14 And someone else:
3 = = 11 This gives to answer for the same math problem. This is bad!!! So we have BEDMAS.What is BEDMAS? It is a trick to remember the order of operations. Brackets, Exponents, Division, Multiplication, Addition, Subtraction. Example: = = = 31 More Examples: ( ) 3 (2 + 3) Special Cases Base 10: what is 10 7? How about 10 n? Base 10 powers are 1 followed by n zeros, where n is the exponent. The first power: What is 5 1? How about ? Any number raised to the exponent of 1 is equal to the base. The power of zero: What is 8 0? How about ? Any nonzero number raised to the exponent of 0 is equal to 1 Now that we have covered the basics of exponents we can look at operations on exponents. Multiplication Since a power is simply a repeated multiplication it would only be natural to have rules for multiplying and dividing powers. How could we simplify : 3
4 2 2 This one is easy it is 2 2 Do you agree that the above could have been written as ? What can we say about the exponents when looking at the following equality? = 2 2 It looks like we are adding the exponents. Consider the multiplication This can be written as Again as This is 8 2 s multiplied together, it is also 2 8. So we get = 2 8 Rule:The multiplication of 2 powers with the same base is simplified to that same base whose exponent is the sum of the 2 exponents: a m a n = a (m+n) NOTE: THE BASES HAVE TO BE THE SAME Examples: Simplify to a single exponent if possible
5 division: Consider This can be written as Again as = = 22 Now with a little work on the fraction we get Here we cancel out some 2 s and we are left with two 2 s Rule: The division of two powers with the same base is simplified to that same base whose exponent is the difference of the 2 exponents: a m a n = a(m n) Knowing this rule, can we now explain why n 0 = 1? Let look at our rule but we are going to let n = m. a m a m = a(m m) What is any number minus that same number? It s zero! What is any number divided by that same number? It s one! this means that our equation above becomes: a 0 = 1 5
6 This is how we can show the property of the exponent zero. Examples: Simplify the following (if the bases are numbers, give their value) = h 45 h 44 Power of a Power What? (3 4 ) 5 is this even legal? Yes, and its not much more than we already covered. Look at (3 4 ) 5 If we consider the inner exponentiation to simply be a number we can write From before we know this to be equal to 3 20 since = 20. We can also see this as 4 5 = 20. A power raised to a power is simplified by multiplying the exponents. (a n ) m = a n m Extended to more than one power, each exponent gets multiplied. (a n b k ) m = a n m b k m Simplify: 1. (5 3 ) 4 2. ( ) (2 3 4) 3 4. (4 3 ) 2 6
7 5. (34 7 ) 8 6. ((6 2 ) 2 ) 2 7. *** ( ) 3 Negative Exponents: Simplify According to our previous rules, this gives 2 2. What does this mean? Looking at this as we did before we see that This can be written as = 2 2 Simplify to get So a negative exponent in the numerator becomes a positive if it is sent to the denominator. Similarly a negative exponent in the denominator becomes a positive exponent in the 1 numerator. That is, 2 2 = 22 Examples: Simplify the following. Write the answers with positive exponents
8 PROBLEMS 1. Write the following as exponents (a) (b) 7 to the (c) (d) (e) (f) (g) Evaluate the following: (a) (b) (c) (d) (e) (f) (g) What is BEDMAS and what does it stand for? BEDMAS is a trick to remember the order of operations and it stands for: bracket, exponent, division, multiplication, addition, subtraction 4. Evaluate (a) (b) (c) (2 4 2) (d) ( ) Simplify if possible (It may help to write down the rules we covered): (a)
9 (b) (c) (d) (e) (f) (g) Simplify if possible: (a) (b) (c) = (d) (e) (f) Simplify if possible: (a) (4 2 ) (b) (3 12 ) = 1 (c) ((4 2 ) 4 ) (d) ( ) (e) ( ) (f) (g) 3 (3 4 ) (6 1 ) If the population of rabbits triples every year, how many rabbits will there be in 5 years if there are currently 2? After one year 2 3, after 2 years 2 3 3, after 3 years , continuing we get = 486 9
10 9. If a bacteria population starts at 100 quadruples every hour, how many bacteria will there be in 6 hours? = = The memory capacity of a computer doubles every year. If you can store 1000 songs on your MP3 player now, how many songs will you be able to store in 10 years? = = You go back in time and tell you parents to buy into Apple. Your parents wisely listen you and invest $1000. Since then, The value of apple has tripled 4 times. how much would your parents $1000 investment be worth now? 1000 tripled 4 times means = 3 4 so your parents 1000 dollar investment would be worth = ** Express 81 7 as a power of base 3. If we look at 81 we notice we can write it as 3 4 since 3 4 = 81 So we can write the above as (3 4 ) 7 From here we can multiply 4 by 7 to get the answer ** Express as a power of base We can simplify this similar to question 11. Here we will change every thing to a power of base 2 then use the rules we learned. (2 4 ) 4 (2 6 ) ** ( ) ** If you have 0 < 10 n < , What is the max value of 3 n? The max value is when n = 0, and therefore 3 n = 1 10
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