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Math 1 – Variable Manipulation Part 2 Student WORKING WITH EXPONENTS Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 53. The "exponent", being 3 in this example, stands for however many times the value is being multiplied. The thing that's being multiplied, being5 in this example, is called the "base". This process of using exponents is called "raising to a power", where the exponent is the "power". The expression "53" is pronounced as "five, raised to the third power" or "five to the third". There are two specially-named powers: "to the second power" is generally pronounced as "squared", and "to the third power" is generally pronounced as "cubed". So "53" is commonly pronounced as "five cubed". When we deal with numbers, we usually just simplify; we'd rather deal with "27" than with "3 3". But with variables, we need the exponents, because we'd rather deal with "x6" than with "xxxxxx". Example: If 3x = 54, then which of the following must be true? a. 1 < x < 2 b. 2 < x < 3 c. 3 < x < 4 d. 4 < x < 5 e. 5 < x Solution: 3x = 54 and the answers are x being between two integers, then we need to know something about x. Since the base is 3, guess that x = 2 so then 32 = 9. Guess x = 3, then 33 = 27. Guess x = 4, then34 = 81. The answer 54 is between 27 and 81 so then x must be between 3 and 4 or C. Sample Question: 1. Solve for x if 63 – 4x -72 = 103

ADD AND SUBTRACT EXPONENTS To add or subtract powers, both the variables and the exponents of the variables must be the same. Perform the required operations on the coefficients, leaving the variable and exponent as they are: x2 + x3 + x3 = x2 + 2x3 Sample Question:

1.

1

1

1

2

3

4

( )2 + ( )2 + ( )2 = ?

MULTIPLY EXPONENTS To multiply powers with the same base, add the exponents: x3 * x4 = x3 + 4 = x7 Example: For all x, the product 3x2 x 5x3 =? a. 8x5 b. 8x6 c. 15x5 d. 15x6 e. 15x8 Math 1 Variable Manipulation Part 2 Student

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Sample Questions: 2. 5x3y5 x 6y2 x 2xy is equivalent to:

3.

The expression −8x3(7x6 − 3x5) is equivalent to:

4.

(2x4y)(3x5y8) is equivalent to:

5.

6.

Which of the following is equivalent to (x)(x)(x)(x3), for all x? a. 4x b. 6x c. x6 d. 4x6 e. 4x4 3 4a x 5a8 = ?

2x2 x 3x2y2 x 5x2y is equivalent to

7.

8.

11y2 11y6 11y8 30y6 30y8

a. b. c. d. e. 9. a. b. c. d. e.

The expression 5y4 * 6y2 is equal to which of the following?

If 8a6b3 < 0, then which of the following CANNOT be true? b0 a=b a0

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DIVIDE EXPONENTS To divide powers with the same base, subtract the exponents: y13 ÷ y8 = y13 - 8 = y5 12𝑧 10

Example: The expression 2 is equivalent to: 4𝑧 a. 3z5 b. 8z5 c. 3z8 d. 8z8 e. 8z12 Solution: Simply the expression by reducing the integers and following the rules for exponents. 12/4 = 3. When dividing like bases, subtract the exponents,

𝑧 10 𝑧2

= z 10 - 2 = z2. The answer is 3z8 or C.

Sample Questions: 10. If xyz ≠ 0, which of the following is equivalent to 𝑥 2𝑦3𝑧 4 (𝑥𝑦𝑧 2 )2

a. b. c. d. e.

?

1/y 1/z y x/yz xyz 𝑛𝑥

11. a. b. c. d. e.

If 𝑛𝑦 = 𝑛2 for all n ≠ 0, which of the following must be true? x+y=2 x–y=2 x×=2 x÷y=2 √𝑥𝑦 = 2 3𝑎4

12.

For all a > 1, the expression 3𝑎6 equals:

13.

For all x > 0, the expression 3𝑥 9 equals:

14.

Reduce 𝑥 4 𝑦 3 𝑧 2 to its simplest terms.

3𝑥 3

𝑥 8 𝑦 12

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RAISE EXPONENTS TO ANOTHER POWER To raise a power to another power, multiply the exponents: (x3)4 = x3 * 4 = x12 To solve for a power of two, take the square root. To solve for a power of three, take the cube root, etc.: x 2 = 16. Take the square root of both sides: √𝑥 2 = √16 becomes x = 4 Example: Which of the following is equivalent to (y3)8? a. y11 b. y24 c. 8y3 d. 8y11 e. 24y Solution: When raising a number with an exponent to another power, you multiply the exponents; therefore, (8y3)8 = 8y3x8 = 8y24 or C Sample Questions: 15. (3x3)3 is equivalent to: a. x b. 9x6 c. 9x9 d. 27x6 e. 27x9 16. Which of the following is equivalent to (4x2)3 ? 8 a. 64x b. 64x6 c. 12x6 b. 12x5 c. 4x6 17. (n5)12 is equivalent to

18.

If nx * n8 = n24 and (n6)y = n18, what is the value of x + y?

ALGEBRA PROBLEMS WITH EXPONENTS 19. What is the value of the expression 2x3 – x2 + 3x + 5 for x = -2?

20.

If f(x) = 2x2 – 6x + 7, then f(-3) = ?

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4x2

21.

For the function h(x) =

– 5x, what is the value of h(–3) ?

22.

A function f(x) is defined as f(x) = −8x2. What is f(−3) ?

23.

If b=a + 3, then (a – b)4 = ?

24.

Let x ∎ y = (x – 2y)2 for all integers x and y. Which of the following is the value of 5 ∎ (-3)?

25. For all real integers, which of the following is ALWAYS an even number? I. x3 + 4 II. 2x + 4 III. 2x2 + 4 a. I only b. II only c. III only d. I and II only e. II and III only 26. a. b. c. d. e.

Which of the following calculations can yield an even integer for any integer a? 2a2 + 3 4a3 + 1 5a2 + 2 6a4 + 6 a6 - 3

ADDING AND SUBTRACTING ROOTS You can add or subtract radical expressions only if the part under the radicals is the same. In other words, treat it like a variable. Just like 2x + 3x would equal 5x:

In other words, you can only add or subtract the numbers in front of the square root sign the numbers under the sign stay the same.

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MULTIPLYING AND DIVIDING ROOTS You can distribute the square root sign over multiplication and division. The product of square roots is equal to the square root of the product: √3 x √5 = √3𝑥5 = √15 The quotient of square roots is equal to the square root of the quotient:

SIMPLIFYING SQUARE ROOTS To simplify a square root, factor out the perfect squares under the radical, square root them, and put the result in front of the part left under the square root sign (the non-perfect- square factors):

Sample Questions: 27. If x is a real number such that x3 = 729, then x2 + √𝑥 = ?

28. a. b. c. d. e.

If x > 1, then which of the following has the LEAST value? √𝑥 √2𝑥 √𝑥 ∗ 𝑥 𝑥 √𝑥 x*x √2100

29.

Which integer is nearest to

30.

If x is a real number such that x3 = 729, then x2 + √𝑥 = ?

31.

√27 + √48 = ?

32.

If √2𝑥 + 5 = 9, then x = ?

Math 1 Variable Manipulation Part 2 Student

√7

?

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33.

What is the smallest integer greater than √58?

34.

If x is a positive real number such that x2 = 16, then x3 + √𝑥 = ?

4

35.

√2

+

2 √3

=?

1 3 5

36. For positive real numbers x, y, and z, which of the following expressions is equivalent to x2 y4 z8 ?

COMPARING EQUATIONS (EQUAL TO) When comparing two problems, set part of the equations to be the same, then set the rest of the equation to be equal and solve. Look at the problem to determine similarities. In many problems, set the base to be the same. Example: If 92x-1 = 33x+3 , then x - ? a. -4 b. -7/4 c. -10/7 d. 2 e. 5 Solution: Express the left side of the equation so that both sides have the same base:

Now that the bases are the same, just set the exponents equal: 4x – 2 = 3x + 3 4x – 3 = 3 + 2 X = 5 or E Other problems will have similar numerators or denominators. Make them exactly the same and set the rest of the equation to be equal. Example: What is the sum of all solutions to a. b. c. d. e.

4𝑥 𝑥−1

=

4𝑥 2𝑥+2

?

-3 -2 2 5 8

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Solution: Since the numerators are both 4x, set the denominators equal to each other x – 1 = 2x + 2 -x – 2 -x - 2 -3 = x or A Sample Questions: 37. In real numbers, what is the solution of the equation 82x + 1 = 41 - x ?

38.

Which real number satisfies (2n)(8) = 163 ?

39.

If 38x = 813x - 2, what is the value of x?

40.

If a, b, and c or consecutive positive integers and 2a x 2b x 2c = 512, then a + b + c = ?

41.

If 𝑦√11 =

4√9

4√9 11

, then y = ?

ADDING AND SUBTRACTING POLYNOMIALS To add or subtract polynomials, combine like terms: (3x2 + 5x - 7) - (x2 + 12) = (3x2 - x2 ) + 5x + (-7 - 12) = 2x2 + 5x - 19 Sample Questions: 42. (6a – 12) – (4a + 4) = ?

43. a. b. c. d. e.

x2 + 60x + 54 – 59x – 82x2 is equivalent to: -26x2 26x6 -81x2 + x + 54 81x2 + x + 54 -83x2 –x -54

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44.

(6a – 12) – (4a + 4) = ?

45.

For all x, x2 – (3x – 2) + 2x(4x – 1) = ?

MULTIPLYING MONOMIALS To multiply monomials, multiply the numbers and the variables separately: 2a x 3a = (2 x 3)(a x a) = 6a2 Sample Questions: 46. Which of the following is an equivalent form of x + x(x + x + x)? a. 5x b. x2 + 3x c. 3x2 + x d. 5x2 e. x3 + x 47. Which of the following is NOT a solution of (x – 5)(x – 3)(x + 3)(x + 9) = 0? a. 5 b. 3 c. -3 d. -5 e. -9 SOLVING "IN TERMS OF" To solve an equation for one variable in terms of another means to isolate the one variable that you are solving for on one side of the equation, leaving an expression containing the other variable on the other side. To solve 3x - 1Oy = -5x + 6y for x in terms of y, isolate x: 3x - 10y = -5x + 6y 3x + 5x = 6y + 1Oy 8x = 16y x= 2y Sample Questions: 48. For all pairs of real numbers M and N where M = 6N + 5, N = ?

49.

If the expression x3 + 2hx - 2 is equal to 6 when x = -2, what is the value of h?

Math 1 Variable Manipulation Part 2 Student

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y)2

50.

What is the value of the expression (x –

51.

How many ordered pairs (x,y) of real numbers will satisfy the equation 5x – 7y = 13? 0 1 2 3 Infinitely many

a. b. c. d. e.

when x = 5 and y = –1 ?

Equivalent Forms Besides plugging in numbers to get an answer, solving in terms of x or another variable, there are ways to simplify equations and express them as equivalent equations. This can be accomplished in many ways. Equations will be equivalent as long as all mathematical rules are followed. Example: Which expression below is equivalent to w(x – (y + z))? a. wx – wy – wz b. wx – wy + wz c. wx – y + z d. wx – y - z e. wxy + wxz Solution: To find an equivalent for the given expression, use the distributive property. First,e valuate the inner parentheses according to the order of operations, or PEMDAS. Distribute the negative sign to (y + z) to get w(x – y – z). Next, distribute the variable w to all terms in parentheses to get wx – wy – wz or A. Choice by fails to distribute the negative sign to the z term. Choices C and D only distribute the w to the first term. Choice E incorrectly distributes wx to the (y + z) term. Sample Questions: 52. Which of the following is a simplified form of 4x – 4y + 3x? a. x(7 – 4y) b. x – y + 3x c. -8xy + 3x d. 7x – 4y e. -4y -x 53.

54.

2𝑟 3

+

4𝑠 5

is equivalent to:

When y = x2, which of the following expressions is equivalent to –y ? a. (–x)2 b. –x2 c. –x d. x–2 e. x

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55. Which of the following is (are) equivalent to the mathematical operation a(b – c) for all real numbers a, b, and c? a. ca – ba b. ab – ac c. (b – c)a a. II only b. I and II only c. I and III only d. II and III only e. I, II and III 56. Which of the following is always equal to y(3 – y) + 5(y – 7)? a. 8y -35 b. 8y -7 c. –y2 + 8y – 7 d. –y2 + 8y – 35 e. 8y3 – 35 57. If W = XYZ, then which of the following is an expression for Z in terms of W, X, and Y? 𝑋𝑌 a. 𝑊 𝑊

b. 𝑋𝑌 c. WXY d. W – XY e. W + XY 58. The expression a[(b – c) + d] is equivalent to: a. ab + ac + ad b. ab – ac + d c. ab – ac + ad d. ab – c + d e. a – c + d UNDERSTANDING CHANGES IN VALUES Another way to evaluate algebraic equations is to look at what happens to the equation as you change the values of the variables. For example, if you increase the value of the denominator, a number will decrease in value. One way to look at these types of questions is to plug in various numbers and see what happens to the answer. Example: Let n equal 3a + 2b -7. What happens to the value of n if the value of a increases by 2 and the value of b decreases by 1? a. It is unchanged. b. It decreases by 1. c. It increases by 4. d. It decreases by 4. e. It decreases by 2. Solution: Set a = 1 and b = 1 and 3a + 2b -7 becomes 3(1) + 2(1) – 7 = 3 + 2 – 7 = -2. Then increase a by 2 and decrease b by 1 and which will mean to set a = 3 and b = 0 and the equation becomes 9 + 0 – 7 = 2. The answer increased by 4. Try a second set of numbers to verify the same result. Set a = 2 and b = 2 to get 3(2) + 2(2) -7 = 6 + 4 – 7 = 3. Increase a by 2 and decrease b by 1 which becomes a = 4 and b = 1. Substituting into the equation to get 12 + 2 – 7 = 7. This answer is also increased by 4. So the answer is c. Math 1 Variable Manipulation Part 2 Student

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Sample Questions: 59. Let x = 2y + 3z -5. What happens to the value of the x if the value of y decreases by 1 and the value of z increases by 2? a. It decreases by 2 b. It is unchanged c. It increases by 1 d. It increases by 2 e. It increases by 4 4 60. In the equation r = (2+𝑘) , k represents a positive integer. As k gets larger without bound, the value of r: f. Gets closer and closer to 4 g. Gets closer and closer to 2 h. Gets closer and closer to 0 i. Remains constant j. Gets larger and larger 61. IF ghjk = 24 and ghkl = 0, which of the following must be true? a. g > 0 b. h > 0 c. j = 0 d. k = 0 e. l = 0 LOGIC Some problems do not require a formula or an equation, they just take some thinking to figure out what the problem is asking and how to solve it. Sample Questions: 62. Fifty high school students were polled to see if they owned a cell phone and an MP3 player. A total of 35 of the students own a cell phone, and a total of 18 of the students own an MP3 player. What is the minimum number of student who own both a cell phone and an MP3 player?

63. Assume that the statements in the below are true.  All students who attend Tarrytown High School have a student ID  Amelia does not attend Tarrytown High School.  Carrie has a student ID.  Tracie has a student ID.  Joseph attends Grayson High school.  Michael is a high school student who attends Tarrytown High School. Considering only these statements, which of the following statements much be true? a. b. c. d. e.

Michal has a student ID. Amelia is not a high school student. Carrie attends Tarrytown High School. Traci attends Tarrytown High School. Joseph does not have a student ID.

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64. If m, n, and p are positive integers such that m + n is even and the value of (m + n)2 + n + p is odd, which of the following must be true? a. b. c. d. e.

m is odd n is even p is odd If n is even, p is odd If p is odd, n is odd

65. Which of the following is true for all consecutive integers m and n such that m < n ? a. m is odd b. n is odd c. n – m is even d. n2 – m2 is odd e. m2 + n2 is even Answer Key

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

61/144 60x4y8 −56x9 + 24x8 6x9y9 x6 20a11 30x6y3 30y6 B y x–y=2 1 𝑎2

x-6 𝑥49 𝑧2

27x9 64x6 n60 19 -21 43 51 -72 81 121 E D 84 √𝑥 17 84 7√3 8 8

Math 1 Variable Manipulation Part 2 Student

34. 66 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65.

4√3+2√2 √6 √𝑥 2 𝑦 6 𝑧 5

8

1

−8 9 2 9 √11 2(a – 8) -81x2 + x + 54 2(a – 8) 9x2 – 5x +2 3x2 + x -5 𝑀−5 6

-4 36 Infinitely many 7x – 4y (10𝑟+12𝑠)

–x2 D D B C E C E 3 A D D

15

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