Exponents Practice Questions

If 2−m=18, what is the value of m?

A) −4
B) −3
C) 1
D) 3
E) 4

Solution:

The correct answer is D.

The equation 2−m=18 is equivalent to 2^m = 8. Since 8 = 23, the value of m must be 3.

What is the value of (−x)2 − (−x) when x = −2?

A) −4
B) −2
C) 0
D) 2
E) 4

Solution:

The correct answer is D.

(−(−2))2 − (−(−2)) = 22 − 2 = 4 − 2 = 2

2x2 × 3x2y2 × 5x2y is equivalent to:

A) 30x^8y^3
B) 30x^8y^2
C) 30x^6y^3
D) 11x^8y^3
E) 11x^6y^2

Solution:

The correct answer is C.

To multiply these quantities, first group like terms:

2x^2 × 3x^2y^2 × 5x^2y = (2 × 3 × 5)(x^2x^2x^2)(y^2y).

To multiply the same base, add the exponents: (2 × 3 × 5)(x^2 x^2 x^2)(y^2y) = 30x^6y^3.

If the value of 4m4n is smaller than 1, which of the following MUST be true?

A) m = 1
B) n = 1
C) m = n
D) m < n
E) m > n

Solution:

The correct answer is D.

The expression 4m4n is equivalent to the expression 4m−n. For this expression to be smaller than 1, the exponent must be negative, so m must be smaller than n.

If x > 0, which of the following expressions will have the smallest value?

A) -2x,x^2,x^2,x^4,x^3
B) −2x
C) x^2
D) x^2
E) x^4
F) x^3

Solution:

The correct answer is A.

Since x is positive, only −2x will be negative, and a negative number is always smaller than a positive number.

If x and y are real numbers such that 0 < x < y2, then which of the following inequalities must be true?

A) x<y
B) y>0y
C) 3x > y2
D) x<1
E) x < 3y2

Solution:

The correct answer is E.

Note that y might be negative. If this is the case, it could still be that y2 is larger than the positive number x, since the square of any negative number is positive, but still less than x and 0 when not squared. This eliminates answer choices A and B. Answer choices C and D will be true for some values but not for others, while answer choice E is always true, since x is already smaller than y2.

If p and q are real numbers such that (pq)2 = 4, then which of the following statements must also be true?

A) pq > 0
B) pq = 2
C) p > 1 and q > 1
D) If p < 0, then q > 0
E) If p > 2, then q < 2

Solution:

The correct answer is E.

It is possible that pq = −2, since −2 × −2 = 4, so answer choices A and B can be eliminated. For answer choice C, it is possible that if the product of two numbers is 2 or −2, one of the numbers is a fraction. For example it could be that p=12 and q=4. Similarly, for answer choice D, p and q could have the same sign or different signs, so this may or may not be true. Finally, in choice E, if p is larger than 2, then you would need to multiply it by something smaller to get a final product of 2. As an example, suppose that p = 3. Since pq = 2, then q=23. This makes answer choice E the only statement that would always be true.

For a real number m, (2m3)4 =

A) 2m^7
B) 2m^12
C) 8m^7
D) 16m^7
E) 16m^12

Solution:

The correct answer is E.

Exponents distribute over multiplication, and when an exponential term is taken to another exponent, you multiply the exponents to simplify the expression. Therefore, (2m^3)4=24m^3×4=16m^12.

Which of the following is equivalent to (14m3)−2 for any real number m?

A) 18^m
B) 18m^5
C) 116m^6
D) 8m^6
E) 16m^6

Solution:

The correct answer is E.

Recall that a−n=1an. Therefore, (14m^3)−2=(4m^3)2=42m^3×2=16m^6.

If the following steps evaluate a function g at the value x, then g(x) =

Step 1: Add 3 to x.

Step 2: Square the result of step 1.

Step 3: Multiply the result of step 2 by 5.

A) (5x + 3)^2
B) 5(x + 3)^2
C) 5(x^2 + 3)
D) (5x)^2 + 3
E) 5x^2 + 3

Solution:

The correct answer is B.

The 3 is added to the x before the term is squared, so the first two steps are (x + 3)^2. In the next step, this term is multiplied by 5.