1. Positive and Negative Exponents Module VI, Lesson 1 Online Algebra VHS@pwcs

2. What is an exponent?  An exponent is a shortcut for writing repeated multiplication.exponent 5454 5 is a base 4 is the exponent, it tells us how many times we multiply the base by itself. = 5 * 5 * 5 * 5 = 625

3. Simplify the following: 1. 4 3 2. 2 6 3. 3 4 4. 2 2 * 5 3 Click to get the answers.  4 * 4 * 4 = 64  2 * 2 * 2 * 2 * 2 * 2 = 64  3 * 3 * 3 * 3 = 81  2 * 2 * 5 * 5 * 5 = 500

4. Do you see any patterns?  Did you notice that when we subtract 1 from the exponent we take away a zero?  What happens when our exponent becomes negative? Another way to think about taking away a zero is to move the decimal point one place to the left. 10 4 10,000 10 3 1,000 10 2 100 10 1 10 10 0 1 10 -1 0.1 10 -2 0.01 10 -3 0.001 10 -4 0.0001

5. Another pattern?  What is the fraction equivalency to each decimal? Notice that:  10 -1 = 1/10 = 1/10 1  10 -2 = 1/100 = 1/10 2  10 -3 = 1/1000 = 1/10 3  10 -4 = 1/10000 = 1/10 4 10 4 10,000 10 3 1,000 10 2 100 10 1 10 10 0 1 10 -1 0.11/10 10 -2 0.011/100 10 -3 0.0011/1000 10 -4 0.00011/10000

6. Zero as an Exponents The first property of exponents is: In algebra: a 0 = 1 In English: any number to the zero power is 1! Try these! 1. 45 0 2. x 0 3. 2r 0  The r is the only thing taken to the 0 power.  So the answer is 2(1) or 2. = 1 = 2

7. Negative Exponents The next property of exponents is:  a -x = or = a x  I like to look at negative exponents this way: First recall that x =, so I think of all numbers/expressions as fractions. the negative exponent is in the numerator, then I switch the number/variable to the denominator. If the negative exponent is in the denominator, then I switch the number/variable to the numerator.

8. Simplify the following:  c -5 We switch the c -5 to the bottom of the fraction, and make it positive. We need something in the numerator, so it is 1, making our answer:

9. Simplify the following  We switch the d -4, to the top of the fraction, leaving a 1 in it’s place:  If we simplify we get: d 4.

10. Simplify these: x -3 y 5 The only variable with a negative exponent is the x, so that is the only exponent we move. Now the things taken to a negative exponent are the a and the b, so these are what we move.

11. Exponent Recap!  An exponent is a short cut to writing repeated multiplication. It tells us how many times we multiply a base by itself.  Any number to the zero power is 1  When dealing with negative exponents, think of all numbers/expressions as fractions. Then switch all variables taken to negative exponent to the opposite side of the fraction bar, making the exponent positive.