Section P.2: Properties of Negative and Zero Exponents. Anything divided by itself is equal to 1. You may also subtract the exponents to get 3 0

Transcription

1 Section P.2: Properties of Negative and Zero Exponents Chapter P Polynomials #-42: Simplify ) a) x2 x 2 Anything divided by itself is equal to. You may also subtract the exponents to get x 0 b) x 0 If you solved part (a) by subtracting the exponents you would get x 2-2 which is equal to x 0. x 0 must equal to the answer to part a. 3) a) Anything divided by itself is equal to. You may also subtract the exponents to get 3 0 b) 3 0 If you solved part (a) by subtracting the exponents you would get which is equal to must equal to the answer to part a. 5) a) 2*3 0 You can check this on your calculator since it only has numbers. By hand the 3 0 is equal to. 2* 2 b) (2*3) 0 Again, you can check this on your calculator. The 2 is in the parenthesis, and this really equals 2 0 *3 0 which is * Solution

2 7) a) -*3 0 If a number is not in a parenthesis, it doesn t turn to when you apply the 0 exponent rule. -* (only the 3 0 turns to a ) - b) (-*3) 0 Since the - is in a parenthesis, the entire -*3 turns to a. 9) a) -3 0 If a negative sign is not in a parenthesis, it doesn t turn to when you apply the 0 exponent rule. Think of this as -*3 0 (only the 3 0 turns to a ) -* - b) (-3) 0 everything is in a parenthesis, this will equal. ) a) -*y 0 - * (only the y 0 turns into a ) - b) (-*y) 0 The negative is in a parenthesis, so the 0 exponent applies to both the - and the y. This is equal to. 3) a) -y 0 This has the same meaning as problem a) that is it equals -*y 0 -*y 0 - * -

3 b) (-y) 0 Anytime a 0 exponent applies to the contents of a parenthesis, the contents of the parenthesis turns into when you apply the exponent. 5) a) 5*6 0 5* 5 b) (5*6) 0 everything is in a parenthesis, this will equal. 7) a) 2x 0 2 * 2 b) (2x) 0 everything is in a parenthesis, this will equal. 9) a) 3xy 0 Only the y 0 turns into a 3x* 3x b) (3xy) 0 everything is in a parenthesis, this will equal 2) a) 3x 0 y 3 * x 0 * y Only the x 0 turns into a. 3 * * y 3y

4 b) (3x 0 y) 0 Since the entire 3x 0 y is in the parenthesis, it all turns to when the 0 exponent rule is applied. 23) x 0 25) ) -3 0 I can simplify this using my calculator. By hand, I need to think of this as -*3 0, and only the 3 0 turns into a. -*3 0 -* - 29) (-2) 0 I can simplify this using my calculator. By hand, since the entire -2 is in a parenthesis the entire problem simplifies to. 3) -w 0 Think of this as -*w 0 -* - 33) (-w) 0 The 0 exponent applies to the content of a parenthesis and the parenthesis simplifies to when you apply the exponent.

5 35) 2c 0 2 * c 0 (only the c 0 turns into a ) 2* 2 37) (2x) 0 The 0 exponent applies to the content of a parenthesis and the parenthesis simplifies to when you apply the exponent. 39) 2bc 0 2b* 2b 4) (2xy 0 ) 0 The 0 exponent applies to the content of a parenthesis and the parenthesis simplifies to when you apply the exponent. 43) a) 3 2 This can be done using a calculator. By hand I need to make the exponent positive by creating a fraction with a in the numerator b) x -2 I need to make the exponent positive by creating a fraction with a in the numerator. x 2 45) a) b) z -3 z 3

6 47) ) x -7 x 7 5) a) 3 2 This can be done using a calculator. By hand I need to make the exponent positive by moving the 3-2 up to the numerator. *3 2 Now order of operations requires me to do the 3 2 first. *9 9 b) x 2 I need to make the exponent positive by moving the x -2 up to the numerator. x 2 or just x 2 53) a) This can be done using a calculator. By hand I need to make the exponent positive by moving the 3-2 up to the numerator. 2*3 2 Now order of operations requires me to do the 3 2 first. 2*9 8 b) 2 x 2 I need to make the exponent positive by moving the x -2 up to the numerator. 2x 2

7 55) 7 5 7*5 7* ) 2 x 5 2x 5 59) x3 x 6 I need to make all of the exponents positive. I can do this by moving the x -6 up to the numerator. My answer will not be a fraction as I will have taken the only thing out of the denominator away from the denominator. x 3 *x 6 x 9 6) x2 x 6 Move the x -6 up x 2 x 6 x 8 63) x3 x 5 x 3 x 5 x 8 65) a) I can simplify this on my calculator. By hand move the 3-2 down to the denominator, and the 2-3 up to the numerator

8 b) x 2 y 3 move the x down and the y up and make the exponents positive. y3 x 2 67) ) x 3 y By hand move the x -3 down to the denominator, and the y - up to the numerator. y x 3 7) x 3 x x x 3 (I can thik of this as: x xxx. Cancel the x from the numerator with one of the x s in the denominator) Really we should just subtract the exponents and leave the two x s in the denominator. x 2 73) x 5 x x x 5 (I can thik of this as: x xxxxx. Cancel the x from the numerator with one of the x s in the denominator) Really we should just subtract the exponents and leave the four x s in the denominator. x 4

9 75) a) I need to move the 3-2 down to the denominator. I will leave a in the numerator as I can t write a fraction without a numerator now I will add the exponents 35 I will use my calculator to simplify b) x 2 x 6 I need to move the x -2 down to the denominator. I will leave a in the numerator as I can t write a fraction without a numerator. x 6 x 2 x 8 77) x 2 x 5 I need to move the x -2 down to the denominator. I will leave a in the numerator as I can t write a fraction without a numerator. x 2 x 5 x 7 79) a) I will move the 3-2 down and the 5-2 up b) 4x 2 y 3 I will move the x -2 down and the y -3 up. 4y3 x 2

10 8) 3x y 3 3y3 x 83) 3x x 3 3x3 x now subtract the exponents, leave the x in the numerator and my answer will not be a fraction. The bigger exponent is in the numerator, so the answer is not a fraction. 3x 2 85) a) I will move the 5-2 up to the numerator. I won t need to write a fraction since I have taken everything away from the denominator. 3*2 2 *5 2 3*4* b) 4x2 y 3 4x 2 y 3 87) a) I will move the 2-2 down to the denominator. Nothing else has a negative exponent. Nothing else moves b) 4x 2 y 3 I will move the x -2 down to the denominator. Nothing else has a negative exponent. Nothing else moves. 4 x 2 y 3

11 89) 2x 3 y Move the x down leave the rest alone. The only negative exponent is the one that applies to the x. 2 x 3 y 9) 2x3 y 5 Move the y up, leave everything else fixed. My answer should not be a fraction as I moved the entire denominator up to the numerator. 2x 3 y 5 93) 6x 2 y 6 2x 5 y 4 Move the x -2, y -6 and y -4 6y 4 2x 5 x 2 y 6 Reduce the 6, then subtract the exponents on the letters. The y winds up in the denominator since the 2 bigger exponent of the y s is in the denominator. The x winds up in the denominator since both x s are in the denominator. Add the exponents of the x s since they are on the same level of the fraction. 2 7x 7 y 2 95) 9x 2 y 6 30x 5 y 4 Only move the x -5 and y -4 9x2 x 5 y 6 y 4 30 Reduce the 9 and add the exponents of the x and y. 30 3x7 y 0 97) 2x 3 x 2 xx x 4

12 99) 2x3 x 5 2x 3 x 5 2x 8 0) a) Subtract the exponents, leave the 2 in the denominator as that is where the larger exponent is. I have to leave in the numerator b) x3 x 5 Subtract the exponents, leave the x in the denominator as that is where the larger exponent is. I have to leave in the numerator. x 5 3 x 2 03) y2 y 6 Subtract the exponents. The y will end up in the denominator as the larger exponent is in the denominator. y 4 05) xy3 x 2 y Subtract the exponents of the x and y. The x winds up in the denominator, the y in the numerator. When you divide with positive exponents you subtract the exponents and leave the letter where the larger exponent starts. y3 x 2 y2 x

13 07) a) 2*3-2 Create a fraction and move the 3-2 to the denominator. Leave the 2 in the numerator since it does not have a negative exponent b) 2x -2 Create a fraction and move the x -2 to the denominator. Leave the 2 in the numerator since it does not have a negative exponent. 2 x 2 09) 5x -3 Create a fraction and move the x -3 to the denominator. Leave the 5 in the numerator since it does not have a negative exponent. 5 x 3 ) 2x - Create a fraction and move the x - to the denominator. Leave the 2 in the numerator since it does not have a negative exponent. 2 x 3) x 3 x -4 Create a fraction and move the x -4 to the denominator. Leave the x 3 in the numerator since it does not have a negative exponent. x 3 x 4 now subtract the exponents and leave the x in the denominator x

14 5) y -2 y 3 Move the y -2 down, but don t move the y 3 y3 y2 now subtract exponents. y 7) a) 3*4 - *5 Move the 4 -, but leave the 3 and the b) 2y 5 x Move the y -5, but leave the 2 and the x. 2x y 5 9) a) 3*2*5-2 Move the 5-2, but leave the 3 and the b) 4xy -3 Move the y -3, but leave the 4 and the x. 4x y 3

15 2) a) 3 - *4*5 - Move both the 3 - and 5 -, leave the b) 4 - xy -3 Move both the 4 - and y -3, leave the x. x 4 y 3 x 4y 3 23) 6xy -4 Only move the y down. 6x y 4 25) 6 - xy -3 Move both the 6 - and y -3, leave the x. x 6 y 3 27) 6x -2 y -3 x 6y 3 Leave the 6, but move the x and y 6 x 2 y 3

16 29) 2 - x -2 y -3 Everything moves to the denominator. I will have to write a in the numerator as I can t write a fraction without a numerator. 2 x 2 y 3 3) 6x -2 x -3 6 x 2 x 3 2x 2 y 3 6 x 5 33) 2 - x -2 x -4 Move everything to the denominator. I will need to leave a in the numerator. 2 x 2 x 4 Now, add the exponents of the x s 2x 6 35) a) ( 2 3 ) 4 Take the reciprocal of the fraction and make the exponent positive. ( 3 2 )4 Now raise the 3 and 2 to the 4 th power

17 b) ( x y ) 4 Take the reciprocal of the fraction and make the exponent positive. ( y x )4 I need to multiply the exponents by 4. I will write the exponents in, but it is not necessary. ( y x )4 y 4 x 4 y4 x 4 37) a) ( ) 2 Take the reciprocal of the fraction and make the exponent positive. ( )2 I can simplify the inside of the parenthesis next. ( 75 2 )2 Now I need to square each number b) ( x 3y 2) 2 Take the reciprocal of the fraction and make the exponent positive. ( 3y2 x )2 32 y 2 2 x 2 9y4 x 2

18 39) a) ( ) 2 I can simplify the inside of the parenthesis first. ( ) 2 ( 3 20 ) 2 Now I can take the reciprocal to make the exponent positive. ( 20 3 ) b) ( 3 x 5 ) 2 I can simplify the inside of the parenthesis first. ( 3 5x ) 2 Now I can take the reciprocal to make the exponent positive. ( 5x 3 )2 52 x x2 9 4) ( 3x 3 y ) I can simplify the inside of the parenthesis first. ( 3 x 3 y ) Now I can take the reciprocal to make the exponent positive. ( x3 y 3 ) x3 y 3

19 43) ( 2x y 4) 2 Simplify the inside of the parenthesis first. (2xy 4 ) 2 Now I can take the reciprocal to make the exponent positive. (2xy 4 ) x 2 y 4 2 4x 2 y 8 45) ( 3x 3 y )2 Simplify the inside of the parenthesis first. ( 3 x 3 y )2 no need to flip this as there are no more negative exponents left. Just square the 3 and multiply the exponents by 2. 9 x 6 y 2 47) ( 2x y 4)3 Simplify the inside of the parenthesis first. (2xy 4 ) 3 no need to flip this as there are no more negative exponents left. Just cube the 2 and multiply the exponents by 3. 8x 3 y 2