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# 8 2power Of Power

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### 8 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>3x4 = 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. Lesson 8.2 Part Two
10. 10. 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
11. 11. Try some <ul><li>(xy )2 = ? </li></ul><ul><li>(xy 2)2 = ? </li></ul><ul><li>(πr 2)4 = ? </li></ul><ul><li>Go to the next slide when you have the solutions to check your work. </li></ul>
12. 12. 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>(π 1 r 2)4 = π 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>
13. 13. Let’s take another look <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>(π 1 r 2)4 = π 4 r 8 </li></ul>
14. 14. Try some more. Use 1 to your advantage when you can. <ul><li>(x 2 y) 3 = (x 2 y1) 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>
15. 15. 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>
16. 16. Powers of -1 <ul><li>Write out (-2) 3 </li></ul><ul><li>(-2)*(-2)*(-2) </li></ul><ul><li>When the exponent is an odd number, the answer can be negative. </li></ul><ul><li>[(-2)*(-2)]*(-2)= </li></ul><ul><li>[+4] * (-2) = -8 </li></ul>
17. 17. Suggestion <ul><li>Once again, the suggestion is to write out the multiplication statements to help you solve tricky exponential products. </li></ul>
18. 18. Simplify <ul><li>(-t) 5 =? </li></ul><ul><li>(-t) 4 =? </li></ul><ul><li>(-5x) 3 =? </li></ul>
19. 19. 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>