1. Page | 1 Olympic College Topic 9 Exponentials Topic 9 Exponentials Definition:In Mathematics an efficient way of representing repeated multiplication is by using the notation a n, such terms are called exponentials where a is called the base and n the exponent. In general we say that a n is ”a to the power of n” but in particular when the exponent is 2 we often use the word squared and when the exponent is 3 we use the word cubed. For example, n terms Here are some examples of exponentials = 125 = 100 5 3 = 10 2 = 2 5 = = 32 (called 5 cubed) (called 10 squared) (called 2 to the power of 5) 1. Properties of Exponentials with the same base. There are three basic operations involving exponentials – they are as follows. Multiplication Rule Division Rule Powers Rule ====== We can give a simple explanation of how these rules work by giving specific examples of each situation. These are not formal proofs but they do give a little insight to why the above rules are true. ==== ==== ====

2. Page | 2 Olympic College Topic 9 Exponentials From the above definitions we can conclude the following properties are also true. a 0 = 1 any positive number to the power of zero is 1. any positive number to the power of – 1 is the same as its reciprocal. any positive number to the power of – n is the same 1 divided by a n. 1 divided by a to a negative power is the same as a to a positive power. Again we can give a simple explanation of how these properties are true. They are not formal proofs but they do give a little insight to how they work. For example,= 7 4-4 =7070 ==1 So 7 0 = 1 = 7 4-5 = == So 7 –1 = ==== ==== ==== 7 – 5 7 4 So = 7 – 5 = 7 4 Example 1: Simplify the following expressions. (The answers are to have only positive exponents). (a) (d) (b) (e) Solution (a): Solution (b): ============ (c) (f) Using the multiplication rule. Using the property that

3. Olympic College Topic 9 Exponentials Page | 3 ==== Solution (c): Solution (d): Solution (e): Using the multiplication rule. Using the property that Using the multiplication rule. ============== =1Using the property that Solution (f):======== Using the multiplication rule. Using the property that Example 2: Simplify the following expressions. (The answers are to have only positive exponents). (a) (d) (b) (e) Solution (a): Solution (b): Solution (c): ================ (c) (f) Using the division rule. Using the property that Using the division rule. Using the property that

4. Olympic College Topic 9 Exponentials Solution (d): Solution (e): Solution (f): ====================== Using the division rule. Using the property that Using the division rule. Using the property that Example 3: Simplify the following expressions. (The answers are to have only positive exponents). (a)(b)(d) Solution (a):= (c) Using the power rule = Solution (b): Solution (c): Solution (d): ============== Using the power rule Using the property that Using the power rule Using the property that Using the power rule = Page | 4

5. Olympic College Topic 9 Exponentials Page | 5 Example 4: Simplify the following expression. (The answers are to have only positive exponents). Solution:Using the multiplication rule====== Using the power rule= Example 5: Simplify the following expression. (The answers are to have only positive exponents). Solution :Using the division rule====== = Example 6: Simplify the following expression Using the power rule Using the property that. (The answers are to have only positive exponents). Solution:================ Using the multiplication rule Using the division rule Using the power rule Using the property that

6. Olympic College Topic 9 Exponentials Page | 6 Exercise 1Simplify the following expressions. (The answers are to have only positive exponents). 1.(a) 1.(d) 2. (a) 2.(d) 3. (a) 3.(d) 4.(a) 5.(a) 6.(a) (b) (e) (b) (e) (b) (e) (b) (c) (f) (c) (f) (c) (f) (c)

7. Olympic College Topic 9 Exponentials 2. Calculations involving multiples of Exponentials with the same base. There are a number of calculations that involve exponentials one such calculation involves multiples of exponentials with the same base. The method used to solve these problems is to separate the multiples and the exponentials and to independently perform the calculations and to then recombine the results. Example 1: Simplify the following expressions. (The answers are to have only positive exponents). (a) (d) (b) (e) Solution (a):= (c) (f) Separate the terms Using the multiplication rule==== 28 Solution (b):=Separate the terms Using the multiplication rule==== 63 Solution (c):=Separate the terms ====== 90 Using the multiplication rule Using the property that Page | 7

8. Olympic College Topic 9 Exponentials Solution (d):=Separate the terms =33Using the multiplication rule 33 Using the property that ======== Solution (e):=Separate the terms =72Using the multiplication rule 72 Using the property that ====== Solution (f):=Separate the terms =28Using the multiplication rule 28 ====== Using the property that = Page | 8

9. Olympic College Topic 9 Exponentials Example 2: Simplify the following expressions. (The answers are to have only positive exponents). (a) (d) (b) (e) Solution (a): Solution (b): Solution (c): ======================== (c) (f) Separate the terms Using the division rule Separate the terms Using the division rule Using the property that Separate the terms Using the division rule Using the property that = Page | 9

10. Olympic College Topic 9 Exponentials Solution (d): Solution (e): Solution (f): Separate the terms Using the division rule Separate the terms Using the division rule Separate the terms Using the division rule ========================== =Using the property that1 = Page | 10

11. Olympic College Topic 9 Exponentials Example 3: Simplify the following expressions. (The answers are to have only positive exponents). (a)(b)(d) Solution (a):==== (c) Separate the terms Using the power rule = Solution (b): Solution (c): Solution (d): ======================== Separate the terms Using the power rule and the property Using the property that Separate the terms Using the power rule Using the property that Separate the terms Using the power rule and the property = Page | 11

12. Olympic College Topic 9 Exponentials Page | 12 Example 4: Simplify the following expression. (The answers are to have only positive exponents). Solution:================ Separate the terms Using the multiplication rule Separate the terms Using the power rule Using the property that = Exercise 2: Simplify the following expressions. (The answers are to have only positive exponents). 1.(a) 1.(d) 2. (a) 2.(d) 3. (a) 3.(d) 4.(a) (b) (e) (b) (e) (b) (e) (b) (c) (f) (c) (f) (c) (f) (c)

13. Olympic College Topic 9 Exponentials 3. Calculations involving Exponentials with the different bases. The method used to solve these problems is very similar to the previous situation, what you do is first separate the multiples and the different exponentials and then independently perform the calculations you then recombine the individual results at the end. Example 1: Simplify the following expressions. (The answers are to have only positive exponents). (a) (c) (e) (b) (d) (f) Solution (a): Solution (b): Solution (c): Separate the terms Using the multiplication rule Separate the terms Using the multiplication rule Using the property that Separate the terms Using the multiplication rule ======================== = Using the property that1 = Page | 13

14. Olympic College Topic 9 Exponentials Solution (d): Solution (e): Solution (f): ========================== Separate the terms Using the multiplication rule Using the property that Separate the terms Using the multiplication rule Using the property that Separate the terms Using the multiplication rule = Page | 14

15. Page | 15 Olympic College Topic 9 Exponentials Example 2: Simplify the following expressions. (The answers are to have only positive exponents). (a) (d) (b) (e) Solution (a): Solution (b): Solution (c): ============================== (c) (f) Separate the terms Using the multiplication rule Using the property that Separate the terms Using the multiplication rule Using the property that Separate the terms Using the multiplication rule Using the property that

16. Olympic College Topic 9 Exponentials ==================== Solution (d): Solution (e): Solution (f): ==== Separate the terms Using the multiplication rule Using the property that Separate the terms Using the multiplication rule Using the property that Separate the terms Using the multiplication rule = Page | 16

17. Olympic College Topic 9 Exponentials Example 3: Simplify the following expressions. (The answers are to have only positive exponents). (a)(c) Solution (a): (b) = (d) Separate the terms Using the power rule = ========== Solution (b): Solution (c): Solution (d): ============== Separate the terms Using the power rule Separate the terms Using the power rule Using the property that Separate the terms Using the power rule and the property Using the property that = Page | 17

18. Olympic College Topic 9 Exponentials Example 4: Simplify the following expression (The answers are to have only positive exponents). Solution:====== Separate the terms Using the multiplication rule Using the property that = Example 5: Simplify the following expression. (The answers are to have only positive exponents). Solution:==================== Using the power Rule Separate the terms Using the multiplication rule Using the property that Page | 18

19. Page | 19 Olympic College Topic 9 Exponentials Exercise 3: Simplify the following expressions (The answers are to have only positive exponents.) 1. (a) (c) (e) (b) (d) (f) 2. 3. (a) (c) (e) (a) (c) (e) (g) (b) (d) (f) (b) (d) (f) (h) 4(a) (c) (e) (g) (b) (d) (f) (h)

20. Page | 20 Olympic College Topic 9 Exponentials Solutions: Exercise 1(b)(c) (d) 1.(a) 2. (a) 3. (a) (b) (c) (e) (d) y 12 (e) (e) x 8 (f) 4.(a) y 18 5.(a) 6.(a) (b) (b) b 8 (c) x 18 (c) Exercise 2: 1.(a) 2. (a) 3.(a) (b) (c) (d) (e) 14x (e) (f) 4.(a)(b)(c) (b) (c) (e) Exercise 3: 1.(a) 2.(a) 3.(a) 3.(e) 4.(a) 4. (e) (b) (f) (b) (f) (d) (c) (g) (c) (g) (f) (d) (h) 1 (d) (h)