1. ExponentsExponents The mathematician’s shorthand

2. Likewise, 5 · 5 · 5 · 5 = 5 4, because there are four 5’s being multiplied together. Power – a number produced by raising a base to an exponent. (the term 2 7 is called a power.) Exponential form – a number written with a base and an exponent. (2 3 ) Exponent – the number that indicates how many times the base is used as a factor. (2 7 ) Base – when a number is being raised to a power, the number being used as a factor. (2 7 )

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

4. Writing exponents 3 · 3 · 3 · 3 · 3 · 3 = 3 6 How many times is 3 used as a factor? (-2)(-2)(-2)(-2) = (-2) 4 How many times is -2 used as a factor? x · x · x · x · x = x 5 How many times is x used as a factor? 12 = 12 1 How many times is 12 used as a factor? 3 6 is read as “3 to the 6 th power.”

5. Evaluating Powers 2 6 = 2 · 2 · 2 · 2 · 2 · 2 = 64 8 3 = 8 · 8 · 8 = 512 5 4 = 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

6. 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 -3 6 mean something entirely different.

7. 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

8. In general, the format for using exponents is: (base) exponent where the exponent tells you how many times the base is being multiplied together. Just a note about zero exponents: powers such as 2 0, 8 0 are all equal to 1. You will learn more about zero powers in properties of exponents and algebra.

9. Simplifying Expressions Containing Powers Simplify 50 – 2(3 · 2 3 ) 50 – 2(3 · 2 3 ) = 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.

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

11. The factors of a power, such as 7 4, can be grouped in different ways. Notice the relationship of the exponents in each product. 7 · 7 · 7 · 7 = 7 4 (7 · 7 · 7) · 7 = 7 3 · 7 1 = 7 4 (7 · 7) · (7 · 7) = 7 2 · 7 2 = 7 4

12. Multiplying Powers with the Same Base To multiply powers with the same base, keep the base and add the exponents. 3 5 · 3 8 = 3 5+8 = 3 13 a m · a n = a m+n

13. Multiply 3 5 · 3 2 = 3 5+2 = 3 7 a 10 · a 10 = a 10+10 = a 20 16 · 16 7 = 16 1+7 = 16 8 6 4 · 4 4 = Cannot combine; the bases are not the same.

14. Dividing Powers with the Same Base To divide powers with the same base, keep the base and subtract the exponents. 6 9 = 6 9-4 = 6 5 6 4 b m = b m-n b n

15. Divide 100 9 = 100 9-3 = 100 6 100 3 x 8 = Cannot combine; the bases are not the same. y 5 When the numerator and denominator of a fraction have the same base and exponent, subtracting the exponents results in a 0 exponent. 1 = 4 2 = 4 2-2 = 4 0 = 1 4 2

16. The zero power of any number except 0 equals 1. 100 0 = 1 (-7) 0 = 1 a 0 = 1 if a ≠ 0

17. How much is a googol? 10 100 Life comes at you fast, doesn’t it?

18. Negative Exponents Extremely small numbers

19. Negative exponents have a special meaning. The rule is as follows: Base negative exponent = Base 1/positive exponent 4 -1 = 1 4 1

20. 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. 10 3 = 10 · 10 · 10 = 1000 10 2 = 10 · 10 = 100 10 1 = 10 = 10 10 0 = 1 = 1 10 -1 = 1/10 10 -2 = 1/10 · 10 = 1/100 10 -3 = 1/10 · 10 · 10 = 1/1000

21. Example: 10 -5 = 1/10 5 = 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 = 1/5 3 = 1/5·5·5 = 1/125

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

23. Evaluate exponents: Get your pencil and calculator ready to solve these expressions. 1)10 -5 = 2)10 5 = 3)(-6) -2 = 4)12 4 /12 6 = 5)12 -3 · 12 6 6)x 9 /x 2 = 7)(-2) -1 = 8)2 3 /2 5 =

24. Problem Solving using exponents The weight of 10 7 dust particles is 1 gram. How many dust particles are in 1 gram? As of 2001, only 10 6 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.

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