GED Math: Powers and Roots
Powers and roots appear in geometry and in comparison questions. The volume of a cube with edge s is s × s × s = s3, and the area of a square with side s is s2. To find a side from the area, take the square root. When comparing powers, change each one to a number (or a decimal) first, and remember that negative exponents and the zero power give values that are fractions or 1. Also follow the order of operations: exponents come before multiplication and addition, so 2 + 32 = 2 + 9 = 11, but (2 + 3)2 = 25.
1. Powers and Exponents
A power is a short way to show repeated multiplication. In 54, the number 5 is the base and the small raised number 4 is the exponent, which tells how many times the base appears as a factor: 54 = 5 × 5 × 5 × 5 = 625. It is read “five to the fourth power.” To evaluate a power, write the base the number of times shown by the exponent and multiply. An exponent is not the same as multiplying the base by the exponent (25 is 32, not 10). When the base is negative, put it in parentheses and count the negative factors: an even exponent gives a positive result and an odd exponent gives a negative result.
Worked Example: Find 25, (−3)2, and (−2)3.
25 = 2 × 2 × 2 × 2 × 2 = 32. (−3)2 = (−3)(−3) = 9 (two negatives). (−2)3 = (−2)(−2)(−2) = −8 (three negatives).
1. Find the value of 24.
Explanation: 2 × 2 × 2 × 2 = 16. (8 is 2 × 4, which is not what an exponent means.)
2. Find the value of 53.
Explanation: 5 × 5 × 5 = 125.
3. Find the value of (−3)2.
Explanation: (−3)(−3) = 9, since two negative factors give a positive.
4. Find the value of (−2)3.
Explanation: (−2)(−2)(−2) = 4(−2) = −8.
5. Find the value of (−2)8.
Explanation: The exponent is even, so the result is positive. 28 = 256.
6. Which expression means 7 × 7 × 7 × 7 × 7?
Explanation: The base 7 is repeated 5 times, so the expression is 75.
2. Special Exponents: 1, 0, and Negative Powers
Exponents can be any number, including 1, 0, and negative numbers, and you should memorize these three rules. First power: a number to the first power is itself. Zero power: any number other than zero raised to the power 0 equals 1. Negative exponent: a number raised to a negative exponent equals a fraction with a numerator of 1 and the positive power in the denominator. A negative exponent does not make the answer negative; it makes it a small fraction (when the base is greater than 1). Also, 1 raised to any power is 1.
| Rule | Example | Value |
|---|---|---|
| Power of 1: a number equals itself | 81 | 8 |
| Power of 0: any nonzero number is 1 | 60 | 1 |
| Negative exponent: 1 over the positive power | 4−2 | 1/42 = 1/16 |
Worked Example: Find 4−2, 250, and 10−3.
4−2 = 1/42 = 1/(4 × 4) = 1/16. 250 = 1. 10−3 = 1/103 = 1/1,000 = 0.001.
1. Find the value of 250.
Explanation: Any nonzero number to the power 0 is 1.
2. Find the value of 41.
Explanation: A number to the first power equals itself.
3. Find the value of 4−2.
Explanation: 4−2 = 1/42 = 1/16.
4. Find the value of 10−3.
Explanation: 10−3 = 1/103 = 1/1,000.
5. Find the value of 115.
Explanation: 1 times itself any number of times is 1.
6. Find the value of (−4)−2.
Explanation: (−4)−2 = 1/(−4)2 = 1/16, since (−4)(−4) = 16.
7. Which has the least value?
Explanation: 3−3 = 1/27 ≈ 0.037, 2−4 = 1/16 = 0.0625, 40 = 1, and 41 = 4. The least is 3−3.
3. Powers on the Calculator
The GED calculator (TI-30XS MultiView) has two power keys. The square key [x²] squares a number: type the number, press [x²], then press enter. The caret key [^] raises a number to any power: type the base, press [^], type the exponent, and press enter. To raise a negative number to a power, put it in parentheses first, using the change-sign key (−) inside the parentheses; otherwise the calculator squares only the number and applies the sign afterward. To enter a negative exponent, use the change-sign key (−) in front of the exponent, not the subtraction key.
| Task | Keys to press | Display |
|---|---|---|
| 242 | 24 [x²] [enter] | 576 |
| 94 | 9 [^] 4 [enter] | 6,561 |
| (−6)2 | ( [(−)] 6 ) [x²] [enter] | 36 |
Worked Example: Find 242, 94, and 1.53 to the nearest tenth.
242 = 576. 94 = 6,561. 1.53 = 1.5 × 1.5 × 1.5 = 3.375, which rounds to 3.4.
1. Use a calculator to find the value of 242.
Explanation: 24 × 24 = 576.
2. Find the value of 94.
Explanation: 9 × 9 × 9 × 9 = 6,561.
3. Find the value of 203.
Explanation: 20 × 20 × 20 = 8,000.
4. Find (3.3)2.
Explanation: 3.3 × 3.3 = 10.89.
5. Find 75.
Explanation: 7 × 7 × 7 × 7 × 7 = 16,807.
6. Find (1.5)3 to the nearest tenth.
Explanation: 1.53 = 3.375, which rounds to 3.4.
7. Why must you put −6 in parentheses on the calculator when finding (−6)2?
Explanation: Without parentheses the calculator treats −62 as the opposite of 62, which is −36. With parentheses it squares −6 and gets 36.
4. Square Roots
To square a number, multiply it by itself: 62 = 36. The square root reverses this. The square root of 36 is 6 because 6 × 6 = 36, and it is written with the radical symbol, √36 = 6. To find a square root, ask, “What number multiplied by itself gives the number under the radical?” Although (−12)(−12) is also 144, on the GED test the square root symbol means the positive root. Numbers like 36, 49, and 144 are perfect squares. For a number that is not a perfect square, such as 90, find the two perfect squares around it to see which whole numbers it is between, or use the calculator: press [2nd], then [x²] (which has √ above it), type the number, and press enter. Press the toggle key to turn the answer into a decimal. Learn the squares of 1 to 15 so you recognize them.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n2 | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 |
Worked Example: Find √144 and approximate √90 to the nearest tenth.
12 × 12 = 144, so √144 = 12. For √90: 9 × 9 = 81 and 10 × 10 = 100, so it is between 9 and 10. The calculator gives 9.4868…, so √90 ≈ 9.5.
1. Find √64.
Explanation: 8 × 8 = 64.
2. Find √121.
Explanation: 11 × 11 = 121.
3. Between which two whole numbers is √40?
Explanation: 6 × 6 = 36 and 7 × 7 = 49, and 40 is between 36 and 49.
4. Between which two whole numbers is √200?
Explanation: 142 = 196 and 152 = 225, and 200 is between them.
5. Use a calculator to find √150 to the nearest tenth.
Explanation: √150 = 12.247…, which rounds to 12.2.
6. Use a calculator to find √57 to the nearest tenth.
Explanation: √57 = 7.5498…. The digit after the tenths place is 4, so round down: 7.5.
7. A square patio has an area of 169 square feet. How long is each side?
Explanation: The side length is √169 = 13 feet, since 13 × 13 = 169.
5. Using Powers and Roots to Solve Problems
Powers and roots appear in geometry and in comparison questions. The volume of a cube with edge s is s × s × s = s3, and the area of a square with side s is s2. To find a side from the area, take the square root. When comparing powers, change each one to a number (or a decimal) first, and remember that negative exponents and the zero power give values that are fractions or 1. Also follow the order of operations: exponents come before multiplication and addition, so 2 + 32 = 2 + 9 = 11, but (2 + 3)2 = 25.
Worked Example: A cube measures 6 inches on each side. Which expression gives its volume in cubic inches, and what is the volume?
Volume = length × width × height = 6 × 6 × 6 = 63 = 216 cubic inches.
1. The cube above has edges of 6 inches. Which expression represents its volume in cubic inches?
Explanation: Volume is length × width × height, three factors of 6, so 63 = 216.
2. What is the volume of a cube with edges 4 inches long?
Explanation: 43 = 4 × 4 × 4 = 64.
3. A square garden has sides 7 feet long. What is its area?
Explanation: Area = 72 = 49.
4. Find the value of 2 + 32.
Explanation: Exponents first: 32 = 9. Then 2 + 9 = 11.
5. Find the value of (2 + 3)2.
Explanation: Parentheses first: 52 = 25.
6. Find the value of 3 × 23.
Explanation: The exponent applies only to 2: 23 = 8, and 3 × 8 = 24.
7. Which is the greatest?
Explanation: 25 = 32, 51 = 5, 100 = 1, and 3−2 = 1/9. The greatest is 25.
Common Mistakes to Avoid
- Multiplying the base by the exponent. 25 means 2 × 2 × 2 × 2 × 2 = 32, not 2 × 5 = 10.
- Thinking a negative exponent makes the answer negative. 4−2 = 1/16, which is positive.
- Saying any number to the power 0 is 0. It is 1 (as long as the base is not zero).
- Forgetting parentheses around a negative base. (−3)2 = 9, but −32 = −9.
- Getting the sign wrong with negative bases. An even exponent gives a positive result and an odd exponent gives a negative one.
- Taking half of the number as the square root. √36 is 6, not 18; ask what number times itself equals 36.
- Doing addition before exponents. In 2 + 32 the power is done first, giving 11.
- Not checking the estimate of a square root. √90 must be between 9 and 10, so an answer such as 45 cannot be right.
Frequently Asked Questions
What does an exponent mean?
An exponent tells how many times the base is used as a factor. In 54, the base 5 appears 4 times: 5 × 5 × 5 × 5 = 625.
What is any number to the power of zero?
Any nonzero number to the power 0 equals 1. For example, 60 = 1 and 1120 = 1.
How do I handle a negative exponent?
Write 1 over the positive power. For example, 4−2 = 1/42 = 1/16. The exponent’s sign flips the base to the denominator; it does not make the answer negative.
How do I estimate a square root without a calculator?
Find the two perfect squares around the number. Since 81 < 90 < 100, √90 is between 9 and 10. Then pick a decimal closer to the nearer square, such as 9.5.
Do I need to memorize square roots for the GED test?
Yes. Knowing the squares from 1 to 15 helps with area, right triangle, and equation problems, and it lets you check calculator answers quickly.
How do I enter a negative number to a power on the GED calculator?
Put the negative number in parentheses, using the change-sign key, then press the square or caret key. For example, ( (−)6 ) [x²] gives 36.
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.