“Three to the Fourth Power”
You may also hear “three raised to the fourth power.” The whole expression is the power.
Free Pre-Algebra Guide
Exponents, Powers, and Square Roots
Learn how exponent notation represents repeated multiplication, how to evaluate powers, and how perfect squares connect to square roots.
Exponents provide a compact way to write repeated multiplication. A complete exponent expression is called a power. Square roots reverse the process of squaring, which makes powers and roots closely related ideas rather than separate rules to memorize.
Base, Exponent, and Power
An exponent expression contains a base and an exponent. The base is the number used as a factor. The exponent is the small raised number that tells how many copies of the base are multiplied together.
4
3 is the base.
It is the repeated factor.
It is the repeated factor.
4 is the exponent.
It tells how many factors of 3 to use.
It tells how many factors of 3 to use.
34 is a power.
Its value is 81.
Its value is 81.
The Exponent Is Not a Factor
In 34, multiply four copies of 3. Do not calculate 3 × 4.
Write Powers as Repeated Multiplication
To expand a power, write the base as a factor exactly as many times as the exponent indicates. Count the factors carefully before multiplying.
Power → Expanded Form
53
= 5 × 5 × 5
Identify the base
The base is 5, so every factor must be 5.
Read the exponent
The exponent is 3, so use three copies of the base.
Write the factors
Write 5 × 5 × 5—not 5 × 3.
Count Factors, Not Multiplication Signs
Three factors contain two multiplication signs. The exponent counts the copies of the base, not the symbols between them.
How to Evaluate a Power
Evaluating a power means finding its numerical value. First expand the power, then multiply one step at a time. Working from left to right keeps the products manageable and makes the reasoning easy to check.
Example: Evaluate 43
- The base is 4 and the exponent is 3.
- Expand: 4 × 4 × 4.
- Multiply the first two factors: 4 × 4 = 16.
- Multiply again: 16 × 4 = 64.
43
= 4 × 4 × 4 = 16 × 4 = 64
A quick estimate can catch mistakes: because
42 = 16,
43 must be four
times 16, or 64.
Squares and Cubes
Powers with exponents 2 and 3 have special names. These names connect exponent notation to geometry.
Squared
A number squared is multiplied by itself once. It can represent the area of a square.
72
= 7 × 7 = 49
Cubed
A number cubed is used as three equal factors. It can represent the volume of a cube.
33
= 3 × 3 × 3 = 27
Powers With Exponents of Zero and One
Exponents of zero and one follow consistent rules. Recognizing them prevents unnecessary calculation.
| Pattern | Rule | Example |
|---|---|---|
| Exponent of 1 | Any number to the first power equals itself. | 121 = 12 |
| Exponent of 0 | Any nonzero number to the zero power equals 1. | 90 = 1 |
| Base of 1 | Every positive whole-number power of 1 equals 1. | 18 = 1 |
Why Does a Zero Exponent Give 1?
Moving down one exponent divides by the base. Since 52 = 25 and 51 = 5, dividing by 5 once more gives 50 = 1.
Perfect Squares
A perfect square is the product of a whole number multiplied by itself. Knowing the common perfect squares makes square roots much faster to recognize.
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
112 = 121
122 = 144
132 = 169
142 = 196
152 = 225
Notice the Pattern
Each square grows by the next odd number: 1, 4, 9, 16, and 25 increase by 3, 5, 7, and 9. This pattern can help you rebuild a forgotten perfect square.
How to Find a Square Root
A square root asks which nonnegative number, multiplied by itself, equals the radicand—the number inside the radical symbol. The symbol √ indicates the principal, or nonnegative, square root.
Ask for the Repeated Factor
= 9 because
92 = 81
Read the radicand
The number inside the radical is 81.
Recall a square fact
9 × 9 = 81.
State the principal root
The nonnegative square root of 81 is 9.
Principal Root Versus Equation Solutions
The expression √81 has one principal value: 9. However, the equation x2 = 81 has two solutions, x = 9 and x = −9, because both numbers square to 81.
Squaring and Square Roots Are Inverse Operations
Squaring starts with a number and produces its perfect square. Taking the principal square root starts with a nonnegative perfect square and returns the original nonnegative number.
82 = 64
Square 8 to get 64.
√64
= 8
Take the square root of 64 to get 8.
Use One Operation to Check the Other
To check √144 = 12, square the proposed answer. Since 122 = 144, the square root is correct.
Worked Exponent and Square-Root Examples
Evaluate 25
- Use five factors of 2.
- Multiply from left to right.
- 2 × 2 × 2 × 2 × 2 = 32.
25 = 32
Rewrite 6 × 6 × 6 × 6
- The repeated factor is 6.
- There are four copies of 6.
- Use base 6 and exponent 4.
6 × 6 × 6 × 6 = 64
Evaluate 140
- The base 14 is nonzero.
- Use the zero-exponent rule.
- Any nonzero base to the zero power is 1.
140 = 1
Find √196
- Recall a factor multiplied by itself.
- 14 × 14 = 196.
- The principal square root is 14.
√196
= 14
Compare 34 and 43
- Evaluate each power separately.
- 34 = 81 and 43 = 64.
- Compare the resulting values.
34 > 43
Is √225 = 15?
- Square the proposed root.
- 15 × 15 = 225.
- The proposed principal root is correct.
152 = 225 ✓
Common Exponent and Square-Root Mistakes
Multiplying the Base by the Exponent
53 means 5 × 5 × 5, which equals 125. It does not mean 5 × 3 = 15.
Using the Exponent as a Factor
In 72, both factors are 7. The expanded form is 7 × 7, not 7 × 2.
Forgetting the Zero-Exponent Rule
A nonzero base to the zero power equals 1, not zero and not the base itself.
Taking Half of the Radicand
A square root is not division by 2. Find the number that squares to the radicand instead.
Giving ± for a Radical Expression
The radical √ indicates the principal, nonnegative root. The ± appears when solving an equation such as x2 = 49.
Assuming a Larger Exponent Always Wins
Both the base and exponent matter. Evaluate the powers before comparing them.
Steps for Solving Powers and Square Roots
1. Identify the operation
Decide whether the expression is a power or a square root.
2. Read the notation
For a power, name the base and exponent. For a root, name the radicand.
3. Build the relationship
Expand the power or recall the matching number multiplied by itself.
4. Apply special rules
Check for exponents of zero or one before doing longer multiplication.
5. Calculate carefully
Multiply one step at a time, or state the principal nonnegative root.
6. Check with the inverse
Expand a power to verify it, or square a proposed square root.
Your answer should match the structure of the notation:
exponents count factors, while square roots identify a
repeated factor.
Practice Exponents, Powers, and Square Roots
Identify bases and exponents, build repeated multiplication, evaluate powers, and match perfect squares with their square roots using guided feedback.
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