Math Links 9 Chapter 2 & Review for Midterm Chapter 2 Highlights: When adding or subtracting fractions, work with parts of the whole that are of equal size (Equivalent fractions). We do this by finding the lowest common denominator or LCD. 2 5 6 5 2 = = ) + = 2) = 8 2 2 2 5 When multiplying 2 proper fractions, you can multiply the numerators and multiply the denominators: 2 2 2 = = = = 2 2 6 7 To divide two fractions, you multiply the reciprocal of the second fraction: 7 2 7 5 5 7 8 = = or or = 0 5 0 2 20 5 5 You can compare fractions with the same denominator by comparing the numerators: 7 6 < because -7<-6 0 0 A perfect square can be expressed as the product of two equal rational factors.6=.9 x.9 = x 2 2 The square root of a perfect square can be determined exactly.6 =. 9 = 9 Multiplying Integers: Dividing Integers: Rules: Rules: ) + + = + ) + + = + 2) + = 2) + = ) + = ) + = ) = + ) = + Chapter Highlights: A power is a short way to represent repeated multiplication Ie. 7 x 7 x 7 = 7 5 x 5 x 5 x 5= A power consists of a base and an exponent. The base represents the number you multiply repeatedly. The exponent represents the number of times you multiply the base Ie. (-) 5 (-) represents the base 5 represents the exponent (-) 5 represents the power Exponent Laws: To multiply powers with the same base, the base stays the same and we add the exponents. a m x a n = a m+n Example: 7 x 2 = 7+2 = 9 5 6 x 5 = To divide powers with the same base, the base stays the same and we subtract the exponents.
a m a n = a m-n Example: 5 8 5 2 = 5 8-2 = 5 6 6 = To simplify a power of a power, the base stays the same and we multiply the exponents. (a m ) n = a mn Example: ( ) 5 = x5 = 20 (6 8 ) = To simplify a power of a product, we distribute the exponent onto every factor of the base. (a x b) m = a m x b m Example: (5 x 6) = 5 x 6 ( x 9) = To simplify a power of a quotient, we distribute the exponent onto both the numerator and the denominator. n n a a 5 5 7 = Example: n = = b b Any base raised to a power of 0 is equal to (as long as the base does not equal 0). a 0 =, a 0 Example: (-0) 0 = but -5 0 = Expressions with powers can have a numerical coefficient. Evaluate the power and then multiply by the coefficient. 7(-2) = 7 x (-8) = -7 2 = Evaluate expressions with powers using the proper order of operations **Remember** BEDMAS Brackets Exponents Divide and multiply in order from left to right Add and subtract in order from left to right Example: 2 8 2 + (- 2 ) = Chapter 2 Practice For # to 5, choose the best answer.. Four students were asked to write the numbers, 2, 0.7, 0.72, and 5 in ascending order. Which 7 student wrote the numbers in the correct order? A Albert : 0.7, 5 7, 2, 0.72, B Beth: 5 7, 0.7,, 2, 0.72 C Carmella: 0.7, 5 7, 0.72, 2, D Devin: 5 7, 0.7, 2, 0.72, 2. Which rational number is between.06 and.07 on a number line? A B 2 C 26 D 08 0 200 25 00 5. Colin was asked to simplify the expression 6. His work is shown below. 8 6 Step 5 5 6 = (6 ) 8 6 8 6 Step 2 20 = 2 2 Step 7 = 2 2 Step 7 = 2 2 In which step did Colin make his first mistake? A Step B Step 2 C Step D Step
. Which rational number is not an example of a square number? A 96 B 0.069 C 9 Complete the statements in #5 to 7. D 20 5. A decimal number, to the nearest tenth, between 2 and 5 6 is. 6. The value of the expression.7.6 ( 2.) +.7 is. Short Answer 7. Determine the value of each of the following to the nearest tenth. a) 0.6 b) 6 8. Write the value of each expression in the form a b. 8 a) b) 25 9 9. Between what two whole numbers does the square root of 2 lie? 0. Determine the number that has a square root of 2... Shavonne is wearing a flat, metal pendant in the shape of a square. The area of the pendant is 0 cm 2. Estimate the dimensions of the pendant. 2. The area of Mara s square pumpkin patch is 2.25 m 2. She has a square tomato garden with the same area. She wants to determine the dimensions of each garden. Mara s solution is shown below. A = s 2 2A = s 2 2(2.25) = s 2.5 = s 2.5 = s 2.2 = s What error did Mara make in her solution? Correct her solution and determine the dimensions of each garden. 5. John created a painting on a large piece of paper with a length of 2 m and a width of m. 8 a) Determine the area of the painting in lowest terms. Express your answer in the form b a c. b) John did not paint to the edges of the paper. He decides that he wants to crop the painting by cutting off m from each of the four sides of the paper. What are the new dimensions of the painting, written in the form a b?
c) What is the area of the cropped painting, in the form a, expressed in lowest terms? b. Match each letter on the number line to one of the following rational numbers. 7 0. 2 5 2. 0.9 5. Which integers are between 6 and 9 2? 6. Evaluate. Show your work. 7 2 b) + c) + 2 0 2 6 d) 2 0 5 f) + 2 g) + 2 7 h) 2 5 6 Chapter Practice For # to 5, select the best answer.. In the equation ( 2) 5 = 2, which number represents the base of the power? A 2 B 2 C D 2 2. Which expression is equivalent to ( 2) ( 2) ( 2) ( 2) ( 2)? A 2 5 B 2 C ( 2) 5 D ( 2) 5. What is the product of 5 2 and 5? A 650 B 25 6 C 5 8 D 5 6
. Devin was asked to simplify the expression 0 2 ( 2 0 ) 2. His work is shown below. 0 2 ( 2 0 ) 2 = 0 6 ( ) 2 Step = 0 6 Step 2 = 0 2 Step = Step In which step did Devin make his first mistake? A Step B Step 2 C Step D Step 5. Two students were asked to write each product of powers as a single power. Their work is shown below. Danica 2 = ( ) ( ) = 5 Frank 2 = 2 = 6 Which of the following statements about their procedures is true? A Frank s procedure contains an error and Danica s does not. B Danica s procedure contains an error and Frank s does not. C Both Danica and Frank have no errors in their procedure. D Both Danica and Frank have errors in their procedure. Complete the statements in #6 and 7. 6. The value of + 0 is. 7. The expression 5 0 Short Answer written as a fraction in simplified form is. 8. Arrange the powers in order from smallest value to largest value. ( ) 2, (2), (), ( ) 5 9. Write each expression as repeated multiplication. a) 7 b) ( 6) 5 c) ( 5) 0. Write each expression as a power in simplified form. a) 6 7 6 b) (2 2 + ) c) ( 2 ). Explain in words the difference between the powers and. 2. For every metre a scuba diver dives below the water surface of a lake, the light intensity is reduced by 5%. The percent of light intensity can be represented by the equation I = 00( 0.05) d, where I is the intensity of light, as a percent, and d is the depth of the dive, in metres. The intensity of light at the surface of the lake is 00%. Austin wanted to determine the light intensity at a depth of m. His solution is shown below. I = 00( 0.05) d I = 00( 0.05) I = 00( 0.05 )
I = 00(0.999875) I 00 Austin realized that it is not possible for the light intensity to be approximately 00% at a depth of m. Explain where Austin made his mistake. a) Correct Austin s mistake and provide a detailed solution to determine the percent of light intensity at a depth of m. Give your answer to the nearest whole percent. b) What is the light intensity at a depth of 5 m? Give your answer to the nearest whole percent.. What is the volume of a cube with a side length of cm? Show your work... A colony of bacteria triples every hour. There are 0 bacteria now. How many will there be after each amount of time? Show your work. a) h b) h c) 2 h d) n h Solutions for Chapter 2 Practice:. A 2. B. A. D 5. 0.7, 0.8 6. 7. 7. a) 0.6 b) 8.0 8. a) 9 5 b) 7 9. and 5 0. 5.29..2 2. Should be A, not 2A. s = 5. a) 9 2 m b) 7 8 m by 5 m c) 85 2 m. A B 2. C 0. D 7 E 0.9 F 2 5. 5,,, 2,, 0,, 2,, 6. b) c) d) 7 f) 7 2 g) 2 h) 5 5 2 0 2 Solutions for Chapter Practice:. D 2. C. D. A 5. D 6. 28 7. 8. (), ( ) 5, 2, ( ) 2 9. a) 8 b) ( ) ( 6) ( 6) ( 6) ( 6) ( 6) c) 5 5 5 0. a) 6 b) 7 c) 2 2. means that a base of is multiplied times: =. means that a base of is multiplied times: = 77 7. 2. a) In the third line, Austin incorrectly distributed the exponent over subtraction to the bases of and 0.05. You can only distribute an exponent over multiplication: (ab) x = a x b x. I = 00(0.95) ; I = 00(0.85775); I 86. The light intensity is approximately 86%. b) When d = 5, I = 6%.. Volume = = 6 cm. a) 0 = 90 b) 0 = 80 c) 0 2 = 5 9 20 d) 0 n 6 Extra Practice: P. 82 8 #5, 6 0 (o.l),,, 6-2, 25 P. 20 2 #6 6, 8-22