Exponents

1. Exponents The mathematician’s shorthand

2. Is there a simpler way to write5 + 5 + 5 + 5?4 · 5 Just as repeated addition can be simplified by multiplication, repeated multiplication can be simplified by using exponents. For example: 2 · 2 · 2 is the same as 2³, since there are three 2’s being multiplied together.

3. Likewise, 5 · 5 · 5 · 5 = 54, because there are four 5’s being multiplied together. Power – a number produced by raising a base to an exponent. (the term 27 is called a power.) Exponential form – a number written with a base and an exponent. (23) Exponent – the number that indicates how many times the base is used as a factor. (27) Base – when a number is being raised to a power, the number being used as a factor. (27)

4. Evaluating exponents is the second step in the order of operations. The sign rules for multiplication still apply.

5. Writing exponents 3 · 3 · 3 · 3 · 3 · 3 = 36How many times is 3 used as a factor? (-2)(-2)(-2)(-2) = (-2)4How many times is -2 used as a factor? x · x · x · x · x = x5How many times is x used as a factor? 12 = 121How many times is 12 used as a factor? 36 is read as “3 to the 6th power.”

6. Evaluating Powers 26 = 2 · 2 · 2 · 2 · 2 · 2 = 64 83 = 8 · 8 · 8 = 512 54 = 5 · 5 · 5 · 5 = 625 Always use parentheses to raise a negative number to a power. (-8)2 = (-8)(-8) = 64 (-5)3 = (-5)(-5)(-5) = -125 (-3)5 = (-3)(-3)(-3)(-3)(-3) = -243

7. When we multiply negative numbers together, we must use parentheses to switch to exponent notation. (-3)(-3)(-3)(-3)(-3)(-3) = (-3)6 = 729 You must be careful with negative signs! (-3)6 and -36mean something entirely different.

8. Note:When dealing with negative numbers, *if the exponent is an even number the answer will be positive.(-3)(-3)(-3)(-3) = (-3)4 = 81*if the exponent is an odd number the answer will be negative.(-3)(-3)(-3)(-3)(-3) = (-3)5 = -243

9. In general, the format for using exponents is:(base)exponentwhere the exponent tells you how many times the base is being multiplied together.Just a note about zero exponents: powers such as 20, 80are all equal to 1. You will learn more about zero powers in properties of exponents and algebra.

10. Simplifying Expressions Containing Powers • Simplify 50 – 2(3 · 23) 50 – 2(3 · 23) = 50 – 2(3 · 8) Evaluate the exponent. = 50 – 2(24) Multiply inside parentheses. = 50 – 48 Multiply from left to right. = 2 Subtract from left to right.

11. Problem Solving Many problems can be solved by using formulas that contain exponents. Solve the problem below: The distance in feet traveled by a falling object is given by the formula d = 16t2, where t is the time in seconds. Find the distance an object falls in 4 seconds.

12. Problem Solving The sum of the first n positive integers is ½(n2 + n). Check the formula for the first 4 positive integers. Then use the formula to find the sum of the first 12 positive integers. 1 + 2 + 3 + 4 = 10 ½(n2 + n) ½(122 + 12) ½(144 + 12) ½(156) 78

13. (3 - 62) = 42 + (3 · 42) 27 + (2 · 52) (-3)5 2(53 + 102) A population of bacteria doubles in size every minute. The number of bacteria after 5 minutes is 15(25). How many bacteria are there after 5 minutes? Simplify and Solve

14. Properties of Exponents Multiplying, dividing powers and zero power.

15. The factors of a power, such as 74, can be grouped in different ways. Notice the relationship of the exponents in each product. 7 · 7 · 7 · 7 = 74 (7 · 7 · 7) · 7 = 73 · 71 = 74 (7 · 7) · (7 · 7) = 72· 72 = 74

16. Multiplying Powers with the Same Base • To multiply powers with the same base, keep the base and add the exponents. • 35 · 38 = 35+8 = 313 • am · an = a m+n

17. Multiply • 35 · 32 = 35+2 = 37 • a10 · a10 = a10+10 = a20 • 16 · 167 = 161+7 = 168 • 64 · 44 = Cannot combine; the bases are not the same.

18. Dividing Powers with the Same Base • To divide powers with the same base, keep the base and subtract the exponents. • 69 = 69-4 = 65 64 • bm = bm-n bn

19. Divide • 1009 = 1009-3 = 1006 1003 • x8 = Cannot combine; the bases are not the same. y5 When the numerator and denominator of a fraction have the same base and exponent, subtracting the exponents results in a 0 exponent. 1 = 42 = 42-2 = 40 = 1 42

20. The zero power of any number except 0 equals 1.1000 = 1(-7)0 = 1a0 = 1 if a ≠ 0

21. How much is a googol?10100Life comes at you fast, doesn’t it?

22. Negative Exponents Extremely small numbers

23. Negative exponents have a special meaning. The rule is as follows: Basenegative exponent = Base1/positiveexponent 4-1 = 1 41

24. Look for a pattern in the table below to extend what you know about exponents. Start with what you know about positive and zero exponents. 103 = 10 · 10 · 10 = 1000 102 = 10 · 10 = 100 101 = 10 = 10 100 = 1 = 1 10-1 = 1/10 10-2 = 1/10 · 10 = 1/100 10-3 = 1/10 · 10 · 10 = 1/1000

25. Example: 10-5 = 1/105 = 1/10·10·10·10·10 = 1/100,000 = 0.00001 So how long is 10-5 meters? 10-5 = 1/100,000 = “one hundred-thousandth of a meter. Negative exponent – a power with a negative exponent equals 1 ÷ that power with a positive exponent. 5-3 = 1/53 = 1/5·5·5 = 1/125

26. Evaluating negative exponents • (-2)-3 = 1/(-2)3 = 1/(-2)(-2)(-2) = -1/8 • 5-3 = 1/53 = 1/(5)(5)(5) = 1/ 125 • (-10)-3 = 1/(-10)3 = 1/(-10)(-10)(-10) = -1/1000 = 0.0001 • 3-4 · 35 = 3-4+5 = 31 = 3 Remember Properties of Exponents: multiply same base you keep the base and add the exponents.

27. Evaluate exponents:Get your pencil and calculator ready to solve these expressions. • 10-5 = • 105 = • (-6)-2 = • 124/126 = • 12-3 · 126 • x9/x2 = • (-2)-1 = • 23/25 =

28. Problem Solving using exponents The weight of 107 dust particles is 1 gram. How many dust particles are in 1 gram? As of 2001, only 106 rural homes in the US had broadband internet access. How many homes had broadband internet access? Atomic clocks measure time in microseconds. A microsecond is 0.000001 second. Write this number using a power of 10.

29. Exponents can be very useful for evaluating expressions, especially if you learn how to use your calculator to work with them.