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GMAT > 5 - Roots and Exponents > Flashcards

5 - Roots and Exponents Flashcards

1
Q

02

A

0

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2
Q

Perfect Cubes 1-10

A
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3
Q

Non-perfect Square Roots to Memorize

A
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4
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A
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5
Q
A
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A
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7
Q
A

NOTE: On the GMAT, it’s not the absolute value when it’s not a variable

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8
Q

Bases of 2

A
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9
Q

Bases of 3

A
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10
Q

Bases of 4

A
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11
Q

Bases of 5

A
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12
Q

Multiplication of Like Bases

A
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13
Q
A
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14
Q

(XA)B

A
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15
Q

12

A

1

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16
Q

22

A

4

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17
Q

32

A

9

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18
Q

42

A

16

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19
Q

52

A

25

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20
Q

62

A

36

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21
Q

72

A

49

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22
Q

82

A

64

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23
Q

92

A

81

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24
Q

102

A

100

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25
Q

112

A

121

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26
Q

122

A

144

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27
Q

132

A

169

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28
Q

142

A

196

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29
Q

152

A

225

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30
Q

22

A

4

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31
Q

23

A

8

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32
Q

24

A

16

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33
Q

25

A

32

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34
Q

26

A

64

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35
Q

27

A

128

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36
Q

28

A

256

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37
Q

29

A

512

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38
Q

210

A

1024

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39
Q

53

A

125

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40
Q

54

A

625

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41
Q

33

A

27

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42
Q

34

A

81

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43
Q

35

A

243

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44
Q

43

A

64

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45
Q

44

A

256

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46
Q

(xa)(ya) =

A

(xy)a

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47
Q

xa/ya =

A

(x/y)a

48
Q
A
49
Q
A
50
Q

Nonzero Based Raised to the Zero Power

A

When a non-zero base is raised to the zero power, the expression equals 1. (e.g., 20 = 1)

51
Q

Any base raised to the first power

A

Any base raised to the first power, the value equals that base. (e.g., 21=2)

52
Q

x-1 =

A

1/x and in general, x-y= 1/xy

Examples:

2-2 = 1/22=1/4

1/33=3-3

(3/7)-3=(7/3)3

53
Q

Adding or subtracting expressions with exponents

210 + 211 + 212

A

When adding or subtracting expressions with exponents, consider factoring out the common factors.

210 + 211 + 212

=> 210(1 + 21 + 22) = 210(7)

54
Q
A
55
Q

2n + 2n =

A

2n + 2n = 2n+1

3n + 3n + 3n = 3n+1

4n + 4n + 4n + 4n = 4n+1

The rule continues on forever with different bases (<u>number of terms must equal base</u>).

56
Q

Exponent Number Properties Case #1

Base: > 1

Exponent: even positive integer

52

A

Exponent Number Properties Case #1

Base: > 1

Exponent: even

=> Result is larger

52 > 5

57
Q

Exponent Number Properties Case #2

Base: > 1

Exponent: odd positive integer > 1

53

A

Exponent Number Properties Case #2

Base: > 1

Exponent: odd positive integer > 1

=>Result is larger

53 > 5

58
Q

Exponent Number Properties Case #3

Base: < -1

Exponent: even positive integer

(-4)2

A

Exponent Number Properties Case #3

Base: < -1

Exponent: even positive integer

=>Result is larger

(-4)2 > 4

59
Q

Exponent Number Properties Case #4

Base: < -1

Exponent: odd positive integer

(-5)3

A

Exponent Number Properties Case #4

Base: < -1

Exponent: odd positive integer

  • => Result is smaller*
  • (5)3 < -5
60
Q

Exponent Number Properties Case #5

Base: positive proper fraction

Exponent: even positive integer

(1/5)2

A

Exponent Number Properties Case #5

Base: positive proper fraction

Exponent: even positive integer

=> Result is smaller

(1/5)2 < (1/5)

61
Q

Exponent Number Properties Case #6

Base: negative proper fraction

Exponent: even positive integer

(-1/5)2

A

Exponent Number Properties Case #6

Base: positive proper fraction

Exponent: even positive integer

=> Result is larger

(-1/5)2 > (-1/5)

(Remember a fraction multiplied by another fraction gets closer to 0 on the number line)

62
Q

Exponent Number Properties Case #7

Base: positive proper fraction

Exponent: odd positive integer > 1

(1/5)3

A

Exponent Number Properties Case #7

Base: positive proper fraction

Exponent: odd positive integer > 1

=> Result is smaller

(1/5)3

63
Q

Exponent Number Properties Case #8

Base: negative proper fraction

Exponent: odd positive integer > 1

(-1/5)3

A

Exponent Number Properties Case #8

Base: negative proper fraction

Exponent: odd positive integer > 1

=> Result is greater

(-1/5)3 > -1/5

(Remember that multiply a fraction by a fraction moves it closer to 0 on the number line.)

64
Q

Exponent Number Properties Case #9

Base: >1

Exponent: positive proper fraction

51/2

A

Exponent Number Properties Case #9

Base: >1

Exponent: positive proper fraction

=> Result is smaller

51/2 < 5

65
Q

Exponent Number Properties Case #10

Base: positive proper fraction

Exponent: positive proper fraction

1/51/2

A

Exponent Number Properties Case #10

Base: positive proper fraction

Exponent: positive proper fraction

=> Result is larger

1/51/2 > 1/5

(1/5 = 0.2, 1/51/2 = 0.45)

66
Q

Square Roots of Large Perfect Squares (Trailing Zeroes)

A
67
Q

Square Roots of Small Perfect Squares (Leading Zeroes)

A
68
Q

Cube Roots of Large Perfect Cubes (Trailing Zeroes)

A
69
Q

Cube Roots of Small Perfect Cubes (Leading Zeroes)

A
70
Q

5x+1 =

A

5x * 5

71
Q

Cube Root of 4 over Cube Root of 4 over Cube Root of 4

A

41/3 * 41/(3)^2 ​* 41/(3)^3

=> 49/27 * 43/27 * 41/27 = 413/27

72
Q

3 Square Roots of .00000256

A
  1. Take the square root of perfect cube and halve the decimals
  2. Repeat for each additional square root symbol

Square root of .00000256 = .0016

=>Square root of .0016 = .04

=>Square root of .04 = .2

(It is similar to a function where perform the first operation and then use the result to enter into the next operation).

73
Q

Factor Exponential Equation

5x - 5x-1 = 500

What is (x-1)2?

A

5x - 5x-1 = 500 and solve for

  1. On the left side, factor to simplest form. => 5x - (5x)(5-1) => 5x(1 - 5-1)

=> 5x(1 - 1/5) =>5x(4/5)

  1. Raise entire equation to eliminate fractions => 5(5x(4/5) = 500)

=> 5x(4) = 500

  1. Factor to get equal bases => 5x22 = 5422 => 5x=54 => x=4
  2. Plug in 4 into the equation (x-1)2 => (x-1)(x-1) => x2 - 2x + 1 => 42 - 2(8) + 1 = 9

(4-1)2 also works because you have a derived value of x, however you would need to factor and solve for the quadratic if you did not have the x value.

74
Q

What is 7/radical(7)?

A

radical(7), it’s the reverse of radical(7)*radical(7)=7

75
Q

Radical(7)/7

A

1/radical(7)

76
Q

What is a tactic for matching the answer choices of exponents?

A

Be aware of what the answer choices are and merge/split as needed

77
Q

Simplify:

numerator: (1/x4 -x2)
denominator: 1-x6

A
  1. Get a common base of x4 in numerator. Note that x6/x4 = x2. numerator: (1/x4 - x6/x4)
    denominator: 1 - x6

=> numerator: (1 -x6/x4)

denominator: (1-x6)
2. Divide numerator and denominator.

numerator 1-x6 * 1

denominator: x4 * 1-x6
3. 1-x6 cancels out. Simplify to get rid of the fraction.

=> 1/x4 = x-4

78
Q

numerator: (1197 * 742 * 572 * 331 * 450)
denominator: k

Which of the following could be the value of k if the result is an integer?

A. 11100 - 1198

B. 745 - 743

C. 574 - 573

D. 334 - 332

E. 299 - 296

A

E. 299 - 296

Because it is less than 450 = (22)50 = 2100. All other values exceed their respective primes.

79
Q

Simplify:

(23p + 23p ​+ 23p ​+ 23p )(33p + 33p​ + 33p​ + 33p​ + 33p​ + 33p​ + 33p​ + 33p​ + 33p​)

A

(23p + 23p ​+ 23p ​+ 23p )(33p + 33p​ + 33p​ + 33p​ + 33p​ + 33p​ + 33p​ + 33p​ + 33p​)

=> 23p(1 + 1 + 1 +1) 33p(1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1)

=> 23p(4) 33p(9)

=> 23p(22) 33p(32)

=> 23p+233p+2

=> 63p+2

80
Q

Radical in the denominator.

A

A radical in the denominator is not considered simplified. You need to “rationalize” the radical.

numerator: 3 + radical(5)
denominator: radical(5)
1. Multiply by radical(5)/radical(5)

=> numerator: 3 + radical(5) (radical(5))

denominator: radical(5)(radical(5))

=> numerator: 3radical(5) + 5

denominator: 5

Done!

81
Q

How do you remove a radical in a binomial?

A

Multiply by the conjugate (change the sign of the binomial).

numerator: 4
denominator: a - radical(b)
1. Multiply by conjugate = a+radical(b)

=> numerator: 4(a+radical(b)

denominator: (a - radical(b))(a+radical(b))

=> numerator: 4a + 4(radical(b))

denominator: a2 - b

82
Q

Exponents in some cases may not be equal even when they appear to be. How can you mitigate this?

A

Test with 0, 1, -1 on data sufficiency questions.

83
Q

Exponents are distributed over which operations?

A

Multiplication and division.

(4abc)2=42a2b2c2

(12/15)2 = 122/152

84
Q

What is the square root and cube root of x in fraction form?

A

Square root of x= x1/2

cube root of x=x1/3

numerator = power of the radicand

denominator = index

Cube root of x5 = x5/3

85
Q

Square root of 5 is approximately?

A

2.2

86
Q

Square root of 6 is approximately?

A

2.4

87
Q

Square root of 7 is approximately?

A

2.6

88
Q

Square root of 8 is approximately?

A

2.8

89
Q

Square root of 2 is approximately?

A

1.4

90
Q

Square root of 3 is approximately?

A

1.7

91
Q

60th root of x = 10th root of 2

Solve for x.

A
  1. Rewrite as fractions.

x1/60 =21/10

  1. Multiply by LCD

60(x1/60 =21/10) => x60/60 = 260/10 => x1 = 26

=> x=64

92
Q

Which is larger 550 or 725?

A

Multiply exponent (only) by 1/gcf.

550(1/25) and 725(1/25) => 52 and 71

550 is larger.

93
Q

Exponential notation can be factored

A

41000 = 4 * 4 * 4 * 499

94
Q

Change (1/2)-2 to a positive exponent and solve.

A

Flip the fraction and change the sign of the exponent.

(2/1)2=4

95
Q

Pemdas for exponents - what is first, distributing the negative sign or the exponent?

-26 - 33

A

Exponent.

-26 - 33

-64 - 27= -91

96
Q

Solving equations with square roots

A
  1. Isolate the radical to one side
  2. square the equation
  3. solve
  4. must check the answer back in the equation
97
Q

Roots in inequalities

A

Roots should generally not be in inequalities unless it is an odd root

98
Q

210 + 210 =

A

211

99
Q

Scientific notation - multiplying co-efficients

(3.5 * 105)(40 * 106)

A

Co-efficients can be multiple or divided separately to make the calculation easier

(3.5 * 105)(40 * 106)

(.35 * 106)(4.0 * 107)

.35*4=1.4

1.4 * 1013

100
Q

Scientific Notation

6 * 106

and

6.4 * 10-6

A

6 * 106= 6,000,000 (add 6 zeroes)

and

6.4 * 10-6=.0000064 (move decimal 6 places to the left

101
Q

Powers can only be used in what two types of inequalities?

A
  1. All terms are positive
  2. When the power is odd (because it won’t change the sign of the term)
102
Q

Factor exponential equation

162x * 324 = 1

A

Factor exponential equation

162x * 324 = 1

Move to common base of 2 (20 = 1)

(24)2x * (25)4 = 20

28x * 220 = 20

8x+20=0

8x=-20

x=-20/8 = -5/2

103
Q

Requirement to multiply radicals

A

must have the same index. If not, convert to a fraction and find the lcd (I think).

104
Q

(1610 - 234)/21n

What is the largest possible value for n?

A

(1610 - 234)/21n

=> (240 - 234)/21n => 234(26-1)/21n=> 234(63)/21n => 234(21x3)/21n

Only one value of 21 so the largest value of n possible is 1.

105
Q

radical(81/1/x)=x

what is x?

A

radical(81/1/x)=x

radical(81*x/1) = x => radical(81x)=x

With the radical on one side and the fraction removed, you can square the equation.

81x=x2

It’s a quadratic (raised above the power of one) so set the equation equal to zero and solve. .

81x - x2 = 0

x(81-x)

x=0, x=81

106
Q

Factor

2x+1

and

2x-1

A

2x+1 = 2x * 21

and

2x-1 = 2x * 2-1

107
Q

Data Sufficiency: Trailing Zeroes

A
108
Q

Complex Roots

A
109
Q

Quadratic Exponents as Expressions

A
110
Q

Trailing Zeroes in Exponenets #2

A
111
Q

Data Sufficiency: if bases are equal, exponents may be equal

A
112
Q

Complex exponent changes

A
113
Q

Complex Roots / “no larger than” integer divisibility

A
114
Q

Data sufficiency 0/1 bullshit

A
115
Q

(7*10^9)(7*10^3)

A

(7*109)(7*103)

49*1013

=>4.9*1014

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