Exponent Practice Problems – Test 2

1. 4⁶/2⁸=

1. 2
2. 8
3. 16
4. 32
5. 64

2. Which of the following expressions is equal to x^mn ?

1. (x^m)ⁿ
2. x^m+n
3. x^mxⁿ
4. nx^m

3. There are 64 squares on a checkerboard. Bobby puts one penny on the first square, two on the second square, four on the third, eight on the fourth, and continues to double the number of coins at each square until he has covered all 64 squares. How many coins must he place upon the last square?

1. a. 2⁶⁴
2. b. 2⁶⁴-1
3. c. 2⁶³
4. d. 2⁶³+1
5. e. 2⁶⁴-2

4. Which of the following expressions is equivalent to ?

1. a. 2x⁸
2. b. x¹⁵
3. c. x²
4. d. x⁸
5. e. 2x¹⁵

5. Simplify the following expression:
50x¹⁸t⁶w³z²⁰ / 5x⁵t²w²z¹⁹

1. 10x¹³t³wz
2. 10x¹³t⁴wz
3. 10x¹²t⁴wz
4. 10x¹³t⁴wz²

6. Simplify the following expression:
(3x² * 7x⁷)+ (2y³ * 9y¹²)

1. 21x¹⁴ + 18y²⁶
2. 10x⁹ + 11y¹⁵
3. 21x¹⁴ + 18y¹⁵
4. 21x⁹ + 18y¹⁵

7. Simplify the following expression:
(2x⁴y⁷m²z) * (5x²y³m⁸)

1. 10x⁶y⁹m¹⁰z
2. 7x⁶y¹⁰m¹⁰z
3. 10x⁵y¹⁰m¹⁰z
4. 10x⁶y¹⁰m¹⁰z

8. If 2⁴ = 4^x, then x = ?

1. 2
2. 4
3. 6
4. 8

9. If 3⁴ = 9^x, then x = ?

1. 2
2. 4
3. 6
4. 8
5. 16

10. If 6^x/(6²+6²+6²)=1/3, then what is the value of x?

1. 2
2. 3
3. 4
4. 5

Answers & Explanations

1. C

Since 4 is the same as 2², 4⁶ = (2²)⁶ = 212 = 2¹². When dividing exponential numbers with the same base, simply subtract the exponent in the denominator from the exponent in the numerator: 2¹²/2⁸=2^12-8=2⁴=16.

2. A

To determine the power of a power, multiply the exponents. This example presents the reverse case: the product of exponents is equivalent to the power of a power. For example 2^2*3=2⁶=64=4³=(2²)³,

3. C

To see this, consider the following table, which shows the numbers of coins added to the first few squares, and the equivalent powers of 2:

The table shows that in this series, the number of coins on each square represents consecutive powers of 2, since the number doubles with each consecutive square. However, the series of powers begins with 0 for the first square, so that for the 64^th square, the number of coins will be 2⁶³.

4. D

In order to multiply two exponential numbers that have the same base, add their exponents. Therefore, x³x⁵=x³⁺⁵=x⁸.
Also note that x³=x*x*x; therefore the expression x³x⁵equals x*x*x*x*x*x*x*x = x⁸.

5. B

To simplify this expression, it is necessary to follow the law of exponents that states:

xⁿ/x^m =x^n-m

First, the 50 can be divided by 5: 50/5 = 10.

Then, it is simply a matter of using the law of exponents described above to simplify the expression:
(50x¹⁸t⁶w³z²⁰)* (5x⁵t²w²z¹⁹)
= 10x^18-5t^6-2w^3-2z ^20-19 
= 10x¹³t⁴wz.

6. D

To simplify this expression, it is necessary to observe the law of exponents that states:

xⁿ x^m = x^n+m

Therefore: (3x² * 7x⁷)+ (2y³ * 9y¹²)
= 37x⁷⁺² + 29y¹²⁺³ 
= 21x⁹ + 18y ¹⁵.

7. D

To simplify this expression, it is necessary to observe the law of exponents that states:

x^m xⁿ = x^m+n

Therefore, (2x⁴y⁷m²z) * (5x²y³m⁸)
= 10x⁴⁺²y⁷⁺³m²⁺⁸z
= 10x⁶y¹⁰m¹⁰z.

8. A

2⁴ = 2 x 2 x 2 x 2 = 16. Therefore, since 4^x = 16, x = 2.

9. A

3⁴ = 3 x 3 x 3 x 3 = 81. Therefore, since 9^x= 81, x = 2.

10. A

The denominator is equal to 3(6²), so that the expression becomes 6^x/(6²+6²+6² )=6^x/(3*6² )=1/3.

If x = 2 so that 6^x = 6², these will cancel on top and bottom, leaving 1/3.