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MATH 8 UNIT 2 EXPONENTS 5. Which expression is equivalent to Test Review a. Multiple Choice 1. Which of the following numbers is an irrational number? A. Hint: Irrational numbers are non-perfect squares, decimals that never end, and the number pi. 11 8 B. 3 C 9 xyz 6 y3 z b. 1 6xy 3 z c. 1 xy 3 z d. x 6 y3 z 4 x 2 yz ? 24 xy 4 z 2 Hint: List out all of the x,y, and z’s for the top and the bottom. Cross out the ones that they have in common. Simplify the whole numbers. D. –3 2. Which list contains only rational numbers? (x-4)(x-7)(x5) A. π, 1.6, 3.4 B. –2, 1, 2 1 C. –4, , 3 D. 5 1 , 0, 2.66 2 3. Simplify y 2 3y 3 a. b. c. d. 3y 5 3y 6 4 y5 4 y6 6. Which of the following is a simplified form of this expression? Hint: Rational numbers are perfect squares, decimals that end, decimals that repeat, whole numbers, and fractions. Hint: List out all of the y’s and all of the whole numbers. Then multiply the whole numbers and add up the y’s to determine what the exponent will be. EX: y4 = y∙y∙y∙y a. x-5 b. x140 c. x5 x 11 d. 1 x6 3 2x y2 7. Simplify: y 24 a. b. x3 3y c. x3 y 3 d. x3 6 y3 6 y y2 a. 3 b. 4 c. y 3 4. Simplify d. y4 Hint: List out all of the y’s and cancel out what they have in common. x3 12 y Hint: List out all of the x’s and cross off what they have in common. Remember, you can not keep negative exponents. You must make it positive and send it to the denominator. Hint: List out all the whole numbers, x’s and y’s next to each other since they are being multiplied. Then cancel out what they have in common. Also, simplify the whole numbers. (m 2n 3 )2 p 3 8. Simplify 4 12. Solve the following equation for m. –3m + 7 = 31 Hint: Multiply the exponents on top by 2. Then Move any negative ones to the opposite (top or bottom) of where they are. List out all the letters, and cancel what they have in common. 1 a. m p n1 b. m4 p6 n6 m4 p3 c. n6 d. m 4n p 9. Calculate a. b. c. d. 3 8 4 16 32 64 . Hint: m = –8 m = –3 m = 24 m = 27 1. Move any number not attached to the variable. 2. Get the variable by itself. 13. Solve the following equation for f. 2(5f – 11) = 28 a. b. c. d. f=3 f=4 f=5 f=7 Hint: 1. Distribute the 2 into the parentheses. 2. Move the number not attached to the variable to the other side. 3. Get the variable by itself. 14. . Solve the following equation for x. Hint: What number times itself 3 times will give us 64? 4.07 108 4.07 10 8 407 10 6 407 10 6 Hint: Move the decimal to the right until the number you get is > 1 and < 10. The exponent will be the amount of spaces the decimal was moved. It will be positive since the standard form number is LARGE. 11. Which shows 0.004 written in scientific notation? a. 0.4 10 2 b. 4 10 3 c. 4 103 2 d. 0.4 10 1 (18x + 12) = –3x + 40 3 a. x = –21 10. One year it was estimated that 407,000,000 soccer balls were sold worldwide. What is this number in scientific notation? a. b. c. d. a. b. c. d. Hint: Move the decimal to the right until the number you get is > 1 and < 10. The exponent will be the amount of spaces the decimal was moved. The exponent will be negative since the standard form number is SMALL. Hint: Distribute the b.. x = 4 c. x = 5 d. x = 12 1 first. 3 This basically means to divide each of the numbers in the parentheses by 3. 15. Solve for x: 4 x 2( x 3) 10 56 a. x 10 b. x 10 c. x 10.5 2 d. x 8 3 Hint: Distribute the 2 into the parentheses. Then combine like terms. Then get the variable by itself. Perform the following operations. SHOW computation (if needed). Write the answer in scientific notation. 16. ( 3.4 105 ) + ( 7.6 105 ) Hint: Write each number in standard form and then add them together. Hint: Multiply the whole 17. ( 5 103 )( 9 10 2 ) numbers. Keep the multiplication sign and the 10. Add the exponents. Write it in standard form. 18. 8 102 4 105 24. Why do we need scientific notation? Give examples from real life and mention the unit of measure which might be used. a. a very small number Hint: Divide the whole numbers. Keep the multiplication sign and the 10. Subtract the exponents. Write it in standard form. 19. (.0721) (2.68 x102 ) 20. (.054)(3.2 x105 ) Hint: Change the 2nd number to standard form. Add the two numbers. Hint: Change the 2nd number to standard form. Multiply the two numbers together. 21. . The answer on the calculator shows 2.9084E-4. What is the answer in standard form? Explain what the calculator does. Hint: The number after the E tells you what the exponent is. 2 22. An earthquake of magnitude 3.0 is 10 times stronger than an earthquake of magnitude 1.0. An earthquake of magnitude 8.0 is 107 times stronger than an earthquake of magnitude 1.0. How many times stronger is an earthquake of magnitude 8.0 than an earthquake of magnitude 3.0? Hint: Compare the exponent amounts. 23. The image of a dust mite from a scanning electron microscope is 1.5 10 2 millimeters wide. The image is 5 102 times life size. How many millimeters wide is the dust mite? Hint: Microscope size Life size AND Simplify! b. a very large number . 25. Is (ab)2 equivalent to ab2? Explain. Hint: Everything in parentheses takes the exponent on the outside. 26. Justify why x 0 1 . You may use examples if needed. Hint: What does an exponent of zero mean? 27. Explain the difference between a rational and an irrational number. Use examples. 28. Examine the following equations. Label each equation with the following. No solutions One solution Infinite solutions Hint: No solution, you are left with a false statement. (Ex: 3 = 4) a. 2x 5 x 2 x 3 One solution is when x = a specific number. b. 2 x 5 x 3 x 3 Infinite solutions is when you are left with a true statement. (Ex: 4 = 4) c. 3x 5 x 3 x 3 29. Cami and Margaret are saving money. Cami starts with $7 and saves $13 each week. Maggie starts with $11 and saves $13 each week. When will they have the same amount of money? Write and solve an equation to mathematically prove your answer. Hint: Write an expression for each of them and set them equal to each other since we want them to be the same. Answers 1. a 2. d 3. a 4. d 5. d 6. d 7. b 8. c 9. b 10. a 11.b 12. a 13. c x2 0 Or students may also show that 2 x and x x2 x x 1 , so x 0 1 . that 2 x xx 27. A rational number can be expressed as a fraction. Any terminating ( .25 ) or repeating decimal number ( 1.3 ) can be expressed as a fraction (see below—convert repeating decimal to a fraction). An irrational number has a non-repeating decimal for which we can only estimate the fraction (see estimation process below). 14. b 15. a 6 16. 1.110 1 17. 4.5 10 18. 2 103 19. 9.89 102 20. 1.7280 104 28. a. Infinite. No matter what the value for x, the 21. .00029084 – the calculator’s way of showing scientific notation 29. 7 + 13x = 11 + 13x; so they will never have the same amount of money 22. 105 23. 0.3mm 24. A very small number might be microscopic bacterial growth—probably measured in millimeters. A very large number might relate to the distance from Earth to the sun or to the most distant star in the universe—probably measured in miles or kilometers—definitely involving scientific notation. 25. a 2b 2 ab 2 2 26. x x x x1 x x0 1 Or students may use numbers. Or students may use the base 10 pattern. equation would still be true. b. No solutions. 5 6 or 0 1 c.One solution. X can have only 1 value in which the equation will be true.