Q4: Exponents, Roots

Question 1

Data Sufficiency:

What is the value of x + 2y ?

(1) x + y = 2
(2) 6x + 12y - 3 = 15

Answer 1

(1) is Insufficient.
(2) is Sufficient.

We don’t always have to know both variables individually, when the question is asking for a combined value.

Watch out for the “C Trap”: when we think we need both statements, but one statement is sufficient.

We can manipulate (2), so that the left side is the same as the question:

Add 3 on both sides: 6x + 12y = 18

Divide by 6 on both sides: x + 2y = 3

Answer 2

Question 3

Answer 3

Question 4

What is 4 squared?

Answer 4

4 squared = 4² = 4 * 4 = 16

4 is the base. 2 is the power.

For a power of 2, we use the word “squared”.

(This is because a square with 4 units on each side has an area of 4²)

Question 5

What is 2 cubed?

Answer 5

2 cubed = 2 * 2 * 2 = 2³ = 8

For a power of 3, we use the word “cubed”.

(This is because a cube with 2 units on each side has an area of 2³)

Question 6

What “Perfect Square” less than 100 has 5 as a prime factor?

Answer 6

25 = 5²

a Perfect Square is the square of an integer

For a perfect square, the prime factors have even exponents:

Examples: 36 = 2² * 3²

100 = 2² * 5²

Question 7

What “Perfect Cube” less than 100 has 3 as a prime factor?

Answer 7

3³ = 27

a Perfect Cube is the cube of an integer

For a perfect cube, the prime factors have powers that are a multiple of 3.

Example: 1728 = 2⁶ * 3³ = (2*2*3)³ = 12³

Question 8

2³ * 2⁴ = 2^?

Answer 8

Multiplying Like Bases:

2³ * 2⁴ = 2⁷

^{Add the powers:}

x^a * x^b = x^a+b

Question 9

x^y-2 * x^3y+5 = x^?

Answer 9

x^y-2 * x^3y+5 = x^4y+3

^{Multiplying Like Bases: add the powers}

x^a * x^b = x^a+b

Question 10

Answer 10

Question 11

Answer 11

Question 12

(x²)³ = x^?

Answer 12

(x²)³ = x⁶

Power to a Power Rule:

(x^{<span>a</span>})^{<span>b</span>} = x^ab

Question 13

0^a = ?

Answer 13

0

0 to any power is 0.

Question 14

1^a = ?

Answer 14

1

1 to any power is 1

Question 15

Answer 15

Question 16

What is 0.3³, as a fraction?

Answer 16

Question 17

Is 0.4³ > 0.4² ?

Why or why not?

Answer 17

No.

A fraction between 0 and 1 gets SMALLER as the power increases.

0. 4³ = 64/1000 = .064
1. 4² = 16/100 = .16

Question 18

Is -(1/2)³ > -1/2 ?

Answer 18

Yes.

A positive fraction between 0 and 1, taken to greater power, gets closer to 0, and therefore smaller.

A negative fraction between -1 and 0, taken to an odd power,

is still negative and gets closer to 0, and therefore greater.

(-1/2)³ = -1/8

-1/8 > -1/2

Question 19

is a/b > (a/b)² ?

(1) a < b
(2) a and b are positive

Answer 19

1) Insufficient. If a = 1 and b = 2, 1/2 > 1/4 –> Yes

If a = -1 and b=2, -1/2 < 1/4 –> No

(Note that a positive fraction between 0 and 1 gets smaller when raised to a higher power, but also we have to account for the possibility of a negative number)

2) Insufficient. If a = 1 and b = 2, 1/2 > 1/4 –> Yes

If a = 2 and b = 1, 2 No

Together: Sufficient. (2) rules out the negative number possibility that made (1) Insufficient

Question 20

Positive or negative?

a. (-1)⁵ ?
b. (-2)³
c. -3² ?
d. -3*(-4)⁷ ?

Answer 20

a. Negative. A negative number taken to an odd power is negative.
b. Negative.
c. Negative. Without the parentheses, we have to take the exponent first, before the - sign.

So, 3² = 9, and -3² = -9

d. Positive. (-4)⁷ is negative, so -3*(-4)⁷ is positive

Question 21

a) What is x^-1 ?
b) What is 2^-3 ?

Answer 21

A negative exponent is the Reciprocal of the positive exponent:

(1 divided by the positive exponent)

Question 22

a) Express using a negative exponent
b) Express using a positive exponent

Answer 22

Question 23

If x^-1 = 5, what is x?

Answer 23

Question 24

Answer 24

Square Root is the inverse of Squaring.

Question 25

Answer 25

Question 26

Answer 26

Question 27

Answer 27

Question 28

Answer 28

Question 29

Answer 29

Question 30

Answer 30