4 2y. TlOWT. Write as a power. (0.06) (0.06) (0.06) (0.06) Write as a power. Identify the term that is multiplied repeatedly.

Transcription

1 4 2y X TlOWT Write as a power. (0.06) (0.06) (0.06) (0.06) Write as a power. (1.7) (1.7) (1.7) Identify the term that is multiplied repeatedly. Identify the term that is multiplied repeatedly. The term (0.06) is repeated several times in the expression. This term is being multiplied by itself. It is the base. --- is the base. Count the number of times the repeated term is in the expression. (0.06) (0.06) (0.06) (0.06) The repeated term is included 4 times. The exponent is 4. Count the number of times the repeated term is in the expression. is the exponent. Write the exponential expression. Put the base within parentheses. Put the exponent outside. (0.06)4 Write the exponential expression. Answer: (0.06) (0.06) (0.06) (0.06) = (0.06)4 Answer: (1.7)(1.7)(1.7) = 2002, Renaissance Learning, Inc. California

2 [ Simplify (-3)3-14 I Simplify~ (_2)4 + (-1)3 Simplify the exponents. Remember that a negative sign stays with a number if the negative sign is included within parentheses. The exponent tells you the number of times to multiply the number = - (5. 5) = - 25 (_ 3)3 = ( ) = = = (-3) Simplify the exponents = (_2)4 = (-1)3 = -4 2 (_2)4 + (_1) Simplify from left to right = -53 Simplify from left to right. Answer: _52 + (~3)3-14 = -53 Answer: (_2)4 + (-1)3 = 2002, Renaissance Learning, Inc. California

3 ( 5) - 3 Simplify. "6 ISimplify. ( ~r Review negative exponents. Negative exponents tell you to take the reciprocal of the number being raised to a power. (ifl = (if (~) - 2 = Gf Review negative exponents. To drop the negative sign on the exponent, how should the number being raised to a power change? Take the reciprocal of the number being raised to a negative power. (~r3 = (~r Take the reciprocal of the number being raised to a negative power. (~) -2= Write the product without the exponent and multiply. (~r = ~. ~. ~ = i~~ Write the product without the exponent and multiply. ( 5) Answer: "6 = 125 Answer: ( 9) - 2 "2 = 2002, Renaissance Learning, Inc. California

4 . Tnrn ISimplify. ( - 8) 1 I~imp\ify (loo)l Review fractional exponents. The denominator of a fractional exponent tells you what root to take. Two in the denominator tells you to take the square root of the number. Three in the denominator tells you to take the cube, or third, root of the number. The numerator of a fractional exponent works the same as a whole number exponent. 3 8?!3 b -3 3!.8bB a - = -ya. = -yo Review fractional exponents. The denominator of a fractional exponent tells you The numerator of a fractional exponent works the same as Write the following as radical expressions. 5 7 /4 = 25 2 = Rewrite the expression as a radical. Rewrite the expression as a radical. 2~ (-8) 3~ ;/(_8)2 3 (100) 2 = Take the root of the number. Take the root of the number = -8 ;';-8 = -2 ;/(_ 8)2 = (_ 2)2 Raise the number to a power. Raise the number to a power. ( - 2) 2 = = 4 2 Answer: (-8)3 = 4 3 Answer: (100) 2 = 2002, Renaissance Learning, Inc. California

5 I ( )5 Simplify I Simplify. ( ) Review properties of exponents. Review properties of exponents. When exponential expressions have the same base, When dividing exponential terms that have the same the exponents can be simplified. When dividing exponential terms, exponents on the same base can be subtracted. base, the exponents. 55 = 5(5-2) = = 5(2-5) = 5-3 = l When a power is raised to a power, ~ ~ ~ When an exponential term is raised to a power, the exponents. multiply the two exponents. (5 3 )5 = 5(3'5) = Simplify the fraction. Simplify the fraction _ '25' '3 9 ' _ 5(5-6) _ 5-1 _ (3-5) -2...L Simplify each individual fraction. 25 = 2 = 2 = 22 ~ _ (8-3) =4 5 55'23.48 _ Raise the fraction to a power. Raise the fraction to a power. (---L)5= 5 55'23'48)5 = ( ),5 ( ( )5 55~1O '22)4 _ = ( 7 10 '3 8 ' Answer: ( )5 = _4 56'25' ' '3 8 '2 5 Answer: ( )4 _ Renaissance Learning. 'Inc. California

6 I TII1111 Rewrite x- s using only positive exponents. Rewrite ~ using only positive exponents. x Review negative exponents. Negative exponents tell you to take the reciprocal of a number. When a variable has a negative exponent, you can change it to positive exponent by changing where it is in a fraction. Review negative exponents. When you switch a term with a negative exponent from the numerator to the denominator of a fraction, the negative sign on the exponent _5_.!2( = 5a a-3 1 5a- 3 = ~ a 3 When you move a term with a negative exponent to the opposite part of the fraction, drop the negative sign. Rewrite the expression using only positive exponents. Rewrite the expression using only positive exponents. x-s x-s 1_ x , Renaissance Learning, Inc. California

7 Ffnrn IMultiply. (3x 3 ys) (7x:/) lmultiply. (5x 4 y) (~7/) Simplify exponents with the same base. The two x-terms can be combined. They are exponential expressions with the same base. The exponents can be added. The two y-terms can be combined, too. x 3 X = (x x. x). x = X(:3 + 1) = X4 Simplify exponents with the same base. X4 x 7 = x(- + --) - x y. y3 = y<- + -) = Y l y2= (y.y.y.y.y)(y.y) = y(s+ 2) = y7 Multiply the whole numbers and write the new monomial. Multiply the whole numbers and write the new monomial. 3 7 = 21 (3x 3 ys) (7x:/) = 21x 4 y 7 5 4= (5x 4 y)(4x 7 y) = , Renaissance Learning, Inc. California

8 4x 2 Y Objective-Multiply algebraic: expressions with fractional oxponent Multiply. ~ 1 ]. x 3 (x 6 + x 4 ) Multiply. x - 5 (X X - 2 ) Use the distributive property. Multiply the term outside the parentheses by the terms inside the parentheses X 3 (X 6 + X 4 ).~ X 3 (X 6 ) + X 3 (X 4 ) Use the distributive property. Multiply the term outside the parentheses by the terms inside the parentheses X 5 (x 10 + x 2) = Combine exponential terms having the same base. 2 1 The x terms in x 3 (x 6) can be combined because the exponents have the same base, x. Combine the terms by adding the fractions. ~ 1 (2 1) ~]. (2 :1) X 3 (x 6) =X 3" + 6 X 3 (x 4) = x 3" + 4 Combine exponential terms having the same base. ~ + 1 ~ + J ± ( 2 1) 5 ( 2 3) 17 X 3" + 6 = X 6 X3"+4 =X12 ~ 1 } ~ 11 Answer: x 3 (x 6 + X 4) = X 6 + X 12 l.l! Answer: X S (x 10 + x 2 ) = Renaissance Learning. Inc. California

9 ETIT'! Simplify. - 24ily2-8X 5 y 8 Simplify. -30X 5 y7 5 x 8 y 3 Simplify exponents with the same bases. The two x-terms can be combined. Since the terms are being divided, the exponents can be subtracted. The two y-terms can be combined, too. 3 x,x...,x x5 =,x...;x. x x (3-5) -2 1 = x = x = x 2 Simplify exponents with the same bases. -x 5 - (-- x 8 - X --) / (---) ::-:r=y = Y' y2 _ ys ~ '-1 y. y. Y ',5'-5' y' y. Y (2-8) =y =y = (; y Divide the whole numbers and write the new monomial. Divide the whole numbers and write the new monomial ily2 _ 3 _~y8 x-y ~6-30xY 5X 8 y , Renaissance Learning, Inc. California

10 Simplify. -IS x y z -:-6x 4 y 7 z 6 Simplify. 3x-4y 5 6 z 18x 7 y - 8z- 9 ~ I Make all exponents positive. To change a negative exponent to a positive exponent, switch it to the other part of the fraction. Make sure to move its base, too. 3x --3 3ys For example, -=-5-3. Y x' In this monomial, move the x- and z-terms in the numerator to the denominator. Move the x-term in the denominator to the numerator. 42x - IS y 4z x4l -6x 4/ z XIS/z6z12 Simplify exponents that have the same base and are being multiplied. Add the exponents: Z6 Z 12 = Z(6 + 12) = Zl8 42x4l -6xlSy7izl2-42x4l -6x IS /z I8 Simplify exponents that have the same base and are being divided. Subtract the exponents: x:s = X(4 - IS) = x- ll x x 42x4y4 42-6x 15 y 7 z 18-6x ll iz l8 = --h L (4-7) -3 1 Y Y 7 =y =y =-3 Make all exponents positive. Which terms in the numerator should be moved to the denominator? Which terms in the denominator should be moved to the numerator? 18x7y - 8 z x Y z Simplify exponents that have the same base and are being multiplied. Simplify exponents that have the same base and are being divided. Divide the whole numbers and write the new monomial. Divide the whole numbers and write the new monomial. ~ -~7-6 42x- 15 lz x- 4 /i xllizl8 2002, Renaissance Learning, Inc. California

11 [ Simplify. (-5b 4 c- 8 d 3) ISimplify. (3b - c 2 d r- 5 3 Raise each term to the power outside the parentheses. Raise each term to the power outside the parentheses. (_ 5b 4 c- 8 d 3 ) -2 = (_ 5) - 2. (b4) - 2. (C- II) - 2. (d 3 ) -2 (3b - 6 C 2 d 5 ) - 3 = --- Simplify powers raised to powers. Remember that you can simplify powers raised to powers by multiplying. (-5) -2. (b 4 ) - 2. (C - 8r-2. (d 3 ) - 2 Simplify powers raised to powers. (-5) -2. b- 8 C 16 d- 6 = (-5) - 2b- 8 C I6 d- 6 Make all exponents positive. Put negative exponents in the denominator of a fraction. I6 16 C (-5) -2b- Il C d- 6 = (-5)2 8 6 b d Make all exponents positive. Simplify any remaining numerical terms. Simplify any remaining numerical terms. (-5)2 = C C 8 6 (-5)2 b 8 d 6 25b d Answer: (-5b 4 c- 8 d 3 )-2 = ~ 25b ll d 6 Answer: (3b - 6 C 2 d 5 ) - 3 = 2U02, Renaissance Learning, Inc. California

12 . (8b 3 c 1 0d - 5 ) - 4 SImplify. (2b6c-4d 3) - 5 C (3b 7 c- 3 r 2 ) - 3 ISimplify. (5b - 3c5d 2) 4 Raise the numerator and the denominator to the power outside the parentheses (8b3c10d ~ 5)-4 8-4b-12c- 40d20 (2b6c- 4d3) -5 = 2-5b - 30C20d - 15 Raise the numerator and the denominator to the power outside the parentheses. (3b 7 c- 3 d- 2 ) - 3 Make all exponents positive. Switch bases with negative exponents to the opposite part of the fraction. 8-4b-12C-40d20 25b30d20d15 (5b-- 3C5d-2) - 4 = Make all exponents positive b - 30C20d b12C20/0 Simplify exponents that have the same base and are being multiplied Add the exponents: d 0d = d(20+ 15) = d c 20 c 40 = C( ) = COO.25b30d20d15 25b30d35 84b12C2DC C b Simplify exponents that have the same base and are being multiplied. Simplify exponents that have the same base and are being divided. b 30 (30-12) 18 Subtract the exponents: b12 = b = b Simplify exponents that have the same base and are being divided. 25b30d35 25b18d b c c Step 5: Answer: Simplify any remaining numerical terms = 4096 = b18d35 b 18 d c c 60 (8b3c10d - 5) - 4 buli~5 (2b 6 c -- 4 d 3 ) -5 = 128c60 Step 5: Simplify any remaining numerical terms. (3b 7 c- 3 r 2 ) - 3 Answer: (5b -3 c 5 d - 2) , Renaissance Learning. Inc. California

13 Write 10,470,000 in scientific notation. Write 3.4 x 10-4 in decimal notation. Write in scientific notation. Write x log in decimal notation. Determine the decimal to use to write a number in scientific notation. Determine the decimal to use to write a number in scientific notation. A number in scientific notation has one whole number with the remaining non-zero digits written after the decimal point. Drop all zeros after the final non-zero digit in numbers greater than zero. Drop all zeros before the first non-zero digit in numbers between 0 and 1. The first non-zero digit is The decimal to use is ~~~~- In 10,470,000, the last non-zero digit when you read from left to right is 7. Keep zeros between non-zero digits. The decimal to use is Determine the exponent on 10. Ask yourself, "How many places would I have to move the decimal point to get it back to its original position?".. 1~ 10,470,000 The decimal will move 7 places. Use 7 as the exponent. Since the original number is greater than 0, the exponent is positive. 10,4 70,000 = x 10 7 Determine the exponent on 10. How many places would you have to move the decimal point to get it back to its original position? ---- Since the original number is than 0, the exponent is = Please turn the card over for the rest of the problem. Please turn the card over for the rest of the problem.

14 Write 10,470,000 in scientific notation. Write 3.4 x 10-4 in decimal notation. Write in scientific notation. Write x 10 ri in decimal notation. Determine the direction the decimal point moves when you change a number in scientific notation back to decimal form. The exponent on 10 in 3.4 x 10-4 is negative. Move the decimal point to the left. When the exponent is positive, move the decimal point to the right Determine the direction the decimal point moves when you change a number in scientific notation back to decimal form. The exponent on 10 in x 10 6 is Move the decimal point to the Move the decimal point the number of places indicated by the exponent. For 3.4 x 10-4, move the decimal point 4 places to the left. Move the decimal point the number of places indicated by the exponent. For x 10 6, move the decimal point 3.4 X 10-4 O,gqq~ x 10-4 = X 10 6 = , Renaissance Learning, Inc. California

15 Simplify and express in scientific notation. (1.4 x 10 4 ) (7.6 x 10-3) (2.5 x 10-2) Simplify and express in scientific notation. 2 3 (7.4 x 10 ) (5.0 X 10 ) (8.0 x 10-4 ) Use the commutative property of multiplication to simplify the numerator. This helps you put like numbers together. (1.4 x 10 4 ) (7.6 x 10-3 ) = (1.4 x 7.6). (10 4 x 10-3 ) Use the commutative property of multiplication to simplify the numerator. Use the product of powers property x 10-3 = 10(4 + -3) = 10 1 Multiply the decimals as you normally would. (1.4 x 7.6) x (10 4 x 10-3 ) = x 10 1 Use the product of powers property. Multiply the decimals as you normally would. Use the quotient of powers property to simplify the fraction. Wi --: = 10( ) = 10 3 Use the quotient of powers property to simplify the fraction. Divide the decimals as you normally would. Divide the decimals as you normally would x 101 = X X 10 Answer: (1.4 X 10 4 ) (7.6 x 10 ~ 3) = X 10 3 (2.5 x 10 2) Answer: 1 (7.4 X 10 2 ) (5.0 x 10: ) (8.0 x 10-4 ) 2002, Renaissance Learning, Inc. California

16 Ernrn Last year a large trucking company delivered about 0.8 million tons of goods at an average value of $25,100 per ton. What was the total value of goods delivered? Express your answer in scientific notation. Ms. Z, a pop singer, released a new CD in November. Sales were 2.7 million. In December, sales decreased to 0.4 million. How many times more sales were made in November than in December? Write numbers in scientific notation. 1 million =1,000, million = 800,000 = 8 X ,100 = 2.51 x 10 4 Write numbers in scientific notation. 2.7 million = 0.4 million = Write the problem to be solved. Describe the problem with smaller numbers to help you determine which operation to use. Suppose the problem is about delivering 2 truckloads of goods worth $500 each. 2 truckloads at $500 each = 2 x $500 = $1000 Use multiplication to solve this problem. (8 x 10 5 ) x (2.51 X 10 4 ) Write the problem to be solved. Which operation is needed? Use what you know about exponents to compute. (8 x 10~ x (2.51 X 10 4 ) = (8 x 2.51) x (10 5 x 10 4 ) = X 10 9 This number is not in scientific notation since the number in front of the decimal is greater than nine x 10 9 = X 1010 Use what you know about exponents to compute. Answer the question. The trucking company delivered goods worth. $2.008 x Answer the question. 2002, Renaissance Learning. Inc. California