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9 2power Of Power

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Multiply and Divide Exponents

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9 2power Of Power

1. 1. Power of a Power Finding powers of numbers with exponents (x m ) n = x mn
2. 2. Simplify <ul><li>(2 3 ) 2 </li></ul><ul><li>This means 2 3 *2 3 </li></ul><ul><li>2 3 *2 3 = (2*2*2)*(2*2*2)=2 6 </li></ul>
3. 3. Simplify <ul><li>(4 2 ) 3 </li></ul><ul><li>This means 4 2 *4 2 *4 2 </li></ul><ul><li>4 2 *4 2 *4 2 = (4*4)*(4*4)*(4*4)=4 6 </li></ul>
4. 4. How does this work? <ul><li>Look again </li></ul><ul><ul><li>(4 2 ) 3 = 4 6 </li></ul></ul><ul><ul><li>(2 3 ) 2 =2 6 </li></ul></ul><ul><li>How do the exponents 2 and 3 relate to the exponent 6? </li></ul>
5. 5. Let’s look at some more <ul><li>(3 3)4 = (3*3*3)*(3*3*3)*(3*3*3)*(3*3*3) </li></ul><ul><li>(3 3)4 =3 ?? </li></ul><ul><li>3 (3x4) = 3 12 </li></ul><ul><li>As you can see (3 3)4 shows 3 multiplied by itself 12 times. </li></ul><ul><li>(3 3)4 = 3 (3*4) =3 12 </li></ul>
6. 6. Let’s try some using the Power of Powers Property <ul><li>The Power of Powers Property states that when you have a number to a certain power raised to another power, you multiply the exponents. </li></ul><ul><li>Examples </li></ul><ul><ul><li>(3 3 ) 4 = 3 12 </li></ul></ul><ul><ul><li>(8 2 ) 5 = 8 10 </li></ul></ul><ul><ul><li>(9 1 ) 4 = 9 4 </li></ul></ul>
7. 7. Try some <ul><li>(2 3)4 = ? </li></ul><ul><li>(10 3)2 = ? </li></ul><ul><li>(p 2)5 = ? </li></ul><ul><li>(x m)3 = ? </li></ul><ul><li>Go to the next slide when you have the solutions to check your work. </li></ul>
8. 8. Power of Powers <ul><li>(2 3)4 = 2 12 </li></ul><ul><li>(10 3)2 = 10 6 </li></ul><ul><li>(p 2)5 = p 10 </li></ul><ul><li>(x m)3 = x 3m </li></ul>
9. 9. Raise a monomial to a power <ul><li>(xy) 2 = xy*xy = x*x*y*y = x 2 y 2 </li></ul><ul><li>(xy 2 ) 2= </li></ul><ul><li>If you get stuck with powers of powers, try writing out the multiplication of numbers and variables. </li></ul>(x*y*y)* (x*y*y) = x*y*y*x*y*y = x*x*y*y*y*y = x 2 y 4
10. 10. Try some <ul><li>(xy) 2 = ? </li></ul><ul><li>(xy 2 ) 2 = ? </li></ul><ul><li>(  r 2 ) 4 = ? </li></ul><ul><li>Go to the next slide when you have the solutions to check your work. </li></ul>
11. 11. Solutions <ul><li>(x 1 y) 2 = x 2 y 2 </li></ul><ul><li>(x 1 y 2 ) 2 = x 2 y 4 </li></ul><ul><li>(  1 r 2 ) 4 =  4 r 8 </li></ul><ul><li>Can you see the power of powers property at work? </li></ul><ul><li>If not, try changing the variables that have no exponent to an exponent of one. </li></ul><ul><li>{Once again, 1 comes in handy!} </li></ul>
12. 12. Let’s take another look <ul><li>(xy) 2 =(x 1 y 1 ) 2 = x 2 y 2 </li></ul><ul><li>(x 1 y 2 ) 2 = x 2 y 4 </li></ul><ul><li>(  1 r 2 ) 4 =  4 r 8 </li></ul>
13. 13. Try some more. Use 1 to your advantage when you can. <ul><li>(x 2 y) 3 = (x 2 y 1 ) 3 = x (2*3) y( 1*3) = x 6 y 3 </li></ul><ul><li>(x 2 y 2 z 2 ) 3 = </li></ul><ul><li>(abcd) n = </li></ul><ul><li>(x 2 y 3 ) 5 = </li></ul>
14. 14. Solutions <ul><li>(x 2 y 2 z 2 ) 3 =x 2*3 y 2*3 z 2*3 =x 6 y 6 z 6 </li></ul><ul><li>(abcd) n =a n b n c n d n </li></ul><ul><li>(x 2 y 3 ) 5 =x 2*5 y 3*5 = x 10 y 15 </li></ul>
15. 15. Powers of -1 <ul><li>Write out (-2) 3 = (-2)*(-2)*(-2) </li></ul><ul><li>When the exponent is an odd number, the answer can be negative. </li></ul>
16. 16. Suggestion <ul><li>Once again, the suggestion is to write out the multiplication statements to help you solve tricky exponential products. </li></ul>
17. 17. Simplify <ul><li>(-t) 5 =? </li></ul><ul><li>(-t) 4 =? </li></ul><ul><li>(-5x) 3 =? </li></ul>
18. 18. solutions <ul><li>(-t) 5 = (-t) * (-t) * (-t) * (-t) * (-t) </li></ul><ul><li>=-t 5 </li></ul><ul><li>(-t) 4 =t 4 </li></ul><ul><li>(-5x) 3 =(-5x) (-5x) (-5x) = </li></ul><ul><li>= -5*-5*-5*x*x*x = -125x 3 </li></ul>
19. 19. Negative and Zero Exponents Integrated II Chapter 9.2
20. 20. Negative Integers do NOT mean negative numbers
21. 21. Numbers to the Zero Power <ul><li>Every number to the Zero Power, such as 5 0 = 1. </li></ul><ul><li>We can use last lesson’s division of powers as a proof. </li></ul>
22. 22. Using division to prove <ul><li>Any number divided by itself equals 1. </li></ul><ul><li>Using the Quotient of Powers Property, the exponents would be subtracted. </li></ul><ul><li>6 5-5 = 6 0 = 1 </li></ul>
23. 23. Negative Exponents <ul><li>Negative Exponents do not mean negative numbers. </li></ul><ul><ul><li>4 -5 = </li></ul></ul><ul><ul><li>3 -2 = </li></ul></ul><ul><ul><li>7 -4 = </li></ul></ul>
24. 24. Solve.
25. 25. Simplify. <ul><li>b 6 *b -2 =b 4 = 1 </li></ul><ul><li>b 4 b 4 </li></ul><ul><li>-3y -2 </li></ul><ul><li>-6p -7 </li></ul><ul><li>8a 4 b 7 c -4 </li></ul><ul><li>3a 6 b -6 c -4 </li></ul>
26. 26. Simplify. <ul><li>- 3y -2 = -3 </li></ul><ul><li>y 2 </li></ul><ul><li>-6p -7 = -6 </li></ul><ul><li>p 7 </li></ul><ul><li>8a 4 b 7 c -4 = 8 a 4-6 b 7--6 c -4--4 = 8 b 13 </li></ul><ul><li>3a 6 b -6 c -4 3 3 a 2 </li></ul><ul><li>** (c 0 =1 which when multiplied is no longer part of the answer. </li></ul>
27. 28. Let’s Divide! Dividing Monomials Focus: Quotient of Powers Rule
28. 29. Quotients of Powers <ul><li>How do I find ? </li></ul><ul><li>a*a*a*a*a*a = </li></ul><ul><li>a*a*a*a </li></ul><ul><li>a 2 </li></ul><ul><li>1 = a 2 </li></ul>a*a*a*a*a*a = a*a*a*a
29. 30. Let’s find a different way <ul><li>In the previous slide, you saw that the result of this fraction was a 2 . </li></ul><ul><li>How do 6 and 4 relate to two? </li></ul>
30. 31. Quotient of Powers Property <ul><li>For all non-zero numbers, subtract the exponent of the denominator from the numerator when the bases are the same. </li></ul><ul><li>4 5-2 = 4 3 </li></ul>
31. 32. Let’s prove it. <ul><li>4 5-2 = 4 3 </li></ul>
32. 33. Try Some.
33. 34. Solutions <ul><li>2 10-5 = 2 5 </li></ul><ul><li>3 10-7 = 3 3 </li></ul><ul><li>5 8-3 = 5 5 </li></ul><ul><li>2 3-2 = 2 1= 2 </li></ul>
34. 35. Try Some with variables. <ul><li>x j-1 </li></ul><ul><li>x a+b-c </li></ul><ul><li>x m+1-1 = x m </li></ul>
35. 36. Fun Fun Fun
36. 37. Fun Fun Fun <ul><li>-2x 2-1 y 5-3 =-2xy 2 </li></ul><ul><li>-40a 4-1 b2c -5 </li></ul><ul><li>= -40a 3 b 2 </li></ul><ul><li>c 5 </li></ul>
37. 38. Remember negative exponents? <ul><li>Any time you have a negative exponent, it must be placed in the denominator. </li></ul><ul><li>C -3 = </li></ul>
38. 39. Try Some