1. Welcome to the Exponential Jungle

2. Warm Ups Find the Quotient: 147  3 = Find the Quotient: 273/3 = Find the Quotient: 450/10 = Find the Product: 4 4 4 = Find the Product: (-2)(-2)(-2) = Find the Product: (10)(10)(10)(10) =

3. Section 4.1: Divisibility and Factors One integer is DIVISIBLE by another if the remainder is 0 when you divide. Divisibility Rules for 2, 5, and 10. An integer is divisible by… 2, if it ends in 0, 2, 4, 6, and 8. 5, if it ends in 0 or 5. 10, if it ends in 0.

4. Examples: Divisibility Rules Is the first number divisible by the second? 567 and 2? 1,015 and 5? 111,120 and 10? 53 and 2? 1,118 and 2?

5. Divisibility Rule for 3 and 9. See if you can pick out the pattern… NumberSum of Digits Is the Sum Divisible by: 3? 9? Is the Number Divisible by: 3? 9? 2822 + 8 + 2 = 12YesNoYesNo 4684 + 6 + 8 = 18Yes 2152 + 1 + 5 = 8No 1,0171 + 0 + 1 + 7 = 9Yes

6. Divisibility Rule for 3 and 9. From the pattern we saw before… An integer is divisible by… 3, if the sum of its digits is divisible by 3. 9, if the sum of its digits is divisible by 9.

7. Example Problems: Is the first number divisible by the second? Explain. 64 by 9? 472 by 3? 174 by 3? 43,542 by 9?

8. Finding Factors One integer is a FACTOR of another integer if it divides that integer with remainder zero. number

9. Finding Factors So find all the different combinations of factors 12? 16? 18? 24? 1(12), 2(6), 3(4): so, 1, 2, 3, 4, 6, and 12 1(16), 2(8), 4(4): so, 1, 2, 4, and 16 1(18), 2(9), 3(6): so, 1, 2, 3, 6, 9, and 18 1(24), 2(12), 3(8), 4(6): so, 1, 2, 3, 4, 6, 8, 12, and 24

10. Word Problem: There are 20 choral students singing at a school concert. Each row of singers much have the same number of students. If there are at least 5 students in each row, what are all the possible arrangements?

11. List the positive factors of each number. 10 21 31 1, 2, 5, 10 1, 3, 7, 21 1, 31

12. Section 4.2: Exponents You can use EXPONENTS to show repeated multiplication. (Just like multiplication can show repeated addition). A POWER has two parts: A BASE: the factor. And an EXPONENT: the number of times the base (or factor) is multiplied by itself.

13. Exponents: Exponential Notation So show me the EXPANDED version of 2 to the 6 th power. Or 2 6. 2 2 2 2 2 2 = 64 = 2 6

14. Exponents…Show me! 1.Exponential Notation. 2.And Value. (The answer). Twelve to the first power. Six to the second power, or six squared. The opposite of the quantity seven to the fourth power. Negative eight to the fifth power.

15. Exponential Notation and Answers Twelve to the first power. 12 1, 12 Six to the second power, or six squared. 6 2, 6 6 = 36 The opposite of the quantity seven to the fourth power. -7 4, - (7 7 7 7) 4 = - (2,401) = -2,401 Negative eight to the fifth power. (-8) 5, (-8)(-8)(-8)(-8)(-8) = -32,768

16. Writing Exponential Notation Remember to include the Negative Sign. (-5)(-5)(-5) = (-5) 3 Rewrite the expression using the commutative and associative properties. -2 a b a a = -2 a a a b = -2a 3 b

17. Exponents and Negative Integers So when you multiply 2 negative numbers…what do you get? A positive number When you multiply 3 negative numbers…what do you get? A negative number

18. Exponents and Negative Numbers When you multiply an EVEN number of negative integers, the answer will be positive. When you multiply an ODD number of negative integers, the answer will be negative.

19. Example Problems Write these in exponential notation… 6 6 6 6 = (-3)(-3)(-3) = 4 y x y = 4y 2 x

20. Word Problem: A microscope can magnify a specimen 10 3 times. How many times is that?

21. Example Problems: Simplify: 6 2 Evaluate: - a 4, for a = 2 Evaluate: (-a) 4, for a = 2

22. Orders of Operations 1.Work inside the grouping symbols 2.Simplify any terms with exponents. 3.Multiply and divide in order from left to right. 4.Add and subtract in order from left to right.

23. Example Problem: 4(3+2) 2 = 1.Grouping Symbols: 3+2 = 5 2.Exponents: 5 2 = 5 5 = 25 3.Multiplying/Dividing: 4 25 = 100. 4.Addition/Subtraction?: Nope So, 100 is the answer.

24. Example Problem: -2x 3 + 4y, for x = -2 and y = 3. Substitute Variables: -2(-2) 3 + 4(3) 1.Grouping Symbols: None 2.Exponents: (-2) 3 = (-2)(-2)(-2) = -8 -2(-8)+ 4(3) 3.Multiplication/Division: 16 + 12 = 28 4.Addition/Subtraction: None So, 28 is the answer.