5 - Roots and Exponents Flashcards

Question 1

Q

0²

Question 2

Q

Perfect Cubes 1-10

Question 3

Q

Non-perfect Square Roots to Memorize

Question 4

Q

Question 5

Q

Question 6

Q

Question 7

Q

Answer

A

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

Question 8

Q

Bases of 2

Question 9

Q

Bases of 3

Question 10

Q

Bases of 4

Question 11

Q

Bases of 5

Question 12

Q

Multiplication of Like Bases

Question 13

Q

Question 14

Q

(X^A)^B

Question 15

Q

1²

Question 16

Q

2²

Question 17

Q

3²

Question 18

Q

4²

Question 19

Q

5²

Question 20

Q

6²

Question 21

Q

7²

Question 22

Q

8²

Question 23

Q

9²

Question 24

Q

10²

Question 25

Q

11²

Question 26

Q

12²

Question 27

Q

13²

Question 28

Q

14²

Question 29

Q

15²

Question 30

Q

2²

Question 31

Q

2³

Question 32

Q

2⁴

Question 33

Q

2⁵

Question 34

Q

2⁶

Question 35

Q

2⁷

Question 36

Q

2⁸

Question 37

Q

2⁹

Question 38

Q

2¹⁰

Question 39

Q

5³

Question 40

Q

5⁴

Question 41

Q

3³

Question 42

Q

3⁴

Question 43

Q

3⁵

Question 44

Q

4³

Question 45

Q

4⁴

Question 46

Q

(x^a)(y^a) =

Question 47

Q

x^a/y^a =

Question 48

Q

Question 49

Q

Question 50

Q

Nonzero Based Raised to the Zero Power

Answer

A

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

Question 51

Q

Any base raised to the first power

Answer

A

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

Question 52

Q

x^-1=

Answer

A

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

Examples:

2^-2 = 1/2²=1/4

1/3³=3^-3

(3/7)^-3=(7/3)³

Question 53

Q

Adding or subtracting expressions with exponents

2¹⁰ + 2¹¹+ 2¹²

Answer

A

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

2¹⁰ + 2¹¹+ 2¹²

=> 2¹⁰(1 + 2¹ + 2²) = 2¹⁰(7)

Question 54

Q

Question 55

Q

2ⁿ + 2ⁿ =

Answer

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>).}

Question 56

Q

Exponent Number Properties Case #1

Base: > 1

Exponent: even positive integer

5²

Answer

A

Exponent Number Properties Case #1

Base: > 1

Exponent: even

=> Result is larger

5²> 5

Question 57

Q

Exponent Number Properties Case #2

Base: > 1

Exponent: odd positive integer > 1

5³

Answer

A

Exponent Number Properties Case #2

Base: > 1

Exponent: odd positive integer > 1

=>Result is larger

5³> 5

Question 58

Q

Exponent Number Properties Case #3

Base: < -1

Exponent: even positive integer

(-4)²

Answer

A

Exponent Number Properties Case #3

Base: < -1

Exponent: even positive integer

=>Result is larger

(-4)² > 4

Question 59

Q

Exponent Number Properties Case #4

Base: < -1

Exponent: odd positive integer

(-5)³

Answer

A

Exponent Number Properties Case #4

Base: < -1

Exponent: odd positive integer

=> Result is smaller*
(5)³ < -5

Question 60

Q

Exponent Number Properties Case #5

Base: positive proper fraction

Exponent: even positive integer

(1/5)²

Answer

A

Exponent Number Properties Case #5

Base: positive proper fraction

Exponent: even positive integer

=> Result is smaller

(1/5)² < (1/5)

Question 61

Q

Exponent Number Properties Case #6

Base: negative proper fraction

Exponent: even positive integer

(-1/5)²

Answer

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)}

Question 62

Q

Exponent Number Properties Case #7

Base: positive proper fraction

Exponent: odd positive integer > 1

(1/5)³

Answer

A

Exponent Number Properties Case #7

Base: positive proper fraction

Exponent: odd positive integer > 1

=> Result is smaller

(1/5)³

Question 63

Q

Exponent Number Properties Case #8

Base: negative proper fraction

Exponent: odd positive integer > 1

(-1/5)³

Answer

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.)

Question 64

Q

Exponent Number Properties Case #9

Base: >1

Exponent: positive proper fraction

5^1/2

Answer

A

Exponent Number Properties Case #9

Base: >1

Exponent: positive proper fraction

=> Result is smaller

5^1/2 < 5

Answer 16

A

Exponent Number Properties Case #10

Base: positive proper fraction

Exponent: positive proper fraction

=> Result is larger

1/5^1/2> 1/5

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

Answer 17

A

4^1/3* 4^{1/(3)^2} * 4^{1/(3)^3}

=> 4^9/27* 4^3/27* 4^1/27= 4^13/27

Answer 18

A

Take the square root of perfect cube and halve the decimals
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).

Answer 19

A

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

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

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

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

=> 5^x(4) = 500

Factor to get equal bases => 5^x2² = 5⁴2²=> 5^x=5⁴ => x=4
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.

Answer 20

A

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

Answer 21

A

1/radical(7)

Answer 22

A

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

Answer 23

A

Get a common base of x⁴in numerator. Note that x⁶/x⁴= x^2. 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

Answer 24

A

E. 2⁹⁹- 2⁹⁶

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

Answer 25

A

(2^3p+ 2^3p+ 2^3p+ 2^3p)(3^3p + 3^3p + 3^3p + 3^3p + 3^3p + 3^3p + 3^3p + 3^3p + 3^3p)

=> 2^3p(1 + 1 + 1 +1) 3^3p(1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1)

=> 2^3p(4) 3^3p(9)

=> 2^3p(2²) 3^3p(3²)

=> 2^3p+23^3p+2

=> 6^3p+2

Answer 26

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!

Answer 27

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

Answer 28

A

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

Answer 29

A

Multiplication and division.

(4abc)²=4²a²b²c²

(12/15)²= 12²/15²

Answer 30

A

Square root of x= x^1/2

cube root of x=x^1/3

numerator = power of the radicand

denominator = index

Cube root of x⁵= x^5/3

Answer 31

A

Rewrite as fractions.

x^1/60=2^1/10

Multiply by LCD

60(x^1/60=2^1/10) => x^{60/60 =}2^60/10=> x¹= 2⁶

=> x=64

Answer 32

A

Multiply exponent (only) by 1/gcf.

5^50(1/25)and 7^25(1/25)=> 5² and 7¹

5⁵⁰is larger.

Answer 33

A

4^{1000 =}4 * 4 * 4 * 4⁹⁹

Answer 34

A

Flip the fraction and change the sign of the exponent.

(2/1)²=4

Answer 35

A

Exponent.

-2⁶- 3³

^{-64 - 27= -91}

Answer 36

A

Isolate the radical to one side
square the equation
solve
must check the answer back in the equation

Answer 37

A

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

Answer 38

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

Answer 39

A

6 * 10⁶= 6,000,000 (add 6 zeroes)

and

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

Answer 40

A

All terms are positive
When the power is odd (because it won’t change the sign of the term)

Answer 41

A

Factor exponential equation

16^2x * 32⁴= 1

Move to common base of 2 (2⁰= 1)

(2⁴)^2x * (2⁵)⁴= 2⁰

2^8x * 2²⁰= 2⁰

^8x+20=0

^8x=-20

^{x=-20/8 = -5/2}

Answer 42

A

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

Answer 43

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.}

Answer 44

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

Answer 45

A

2^x+1= 2^x * 2¹

^and

2^x-1 = 2^x* 2^-1

Answer 46

A

(7*10⁹)(7*10³)

49*10¹³

=>4.9*10¹⁴