Exponents are a short way to write repeated multiplication. Instead of writing 2 × 2 × 2 × 2 × 2 × 2, we can write 26. Here 2 is called the base and 6 is called the exponent or power. The expression 26 is read as “two raised to the power six” or “two to the sixth power”.
Exponents are used in arithmetic, algebra, science, computer studies, banking calculations, population growth, compound interest, area, volume, square roots, cube roots, and scientific notation. A very large or very small number becomes easier to write when exponents are used. For example, 1000000 can be written as 106, and 0.001 can be written as 10-3.
This lesson explains exponents in simple language. You will learn the meaning of base and exponent, laws of exponents, positive exponents, zero exponents, negative exponents, fractional exponents, powers of 10, scientific notation, squares, cubes, roots, solved examples, common mistakes, and practice questions.
What Is an Exponent?
An exponent tells how many times the base is multiplied by itself. In an, a is the base and n is the exponent. If n is a positive whole number, then an means a is multiplied by itself n times.
Example: 53 = 5 × 5 × 5 = 125.
Here 5 is the base and 3 is the exponent. The exponent 3 tells us to use 5 three times in multiplication.
Example: 72 = 7 × 7 = 49.
Here 7 is the base and 2 is the exponent. When the exponent is 2, we also call it a square.
Example: 44 = 4 × 4 × 4 × 4 = 256.
Parts of an Exponential Expression
| Expression | Base | Exponent | Meaning | Value |
|---|---|---|---|---|
| 25 | 2 | 5 | 2 × 2 × 2 × 2 × 2 | 32 |
| 34 | 3 | 4 | 3 × 3 × 3 × 3 | 81 |
| 103 | 10 | 3 | 10 × 10 × 10 | 1000 |
| 62 | 6 | 2 | 6 × 6 | 36 |
How to Read Exponents
23 is read as two raised to the power three, two to the third power, or two cubed.
52 is read as five raised to the power two, five to the second power, or five squared.
106 is read as ten raised to the power six.
In school mathematics, the words exponent, power, and index are often used for the same idea. In many Indian school books, “indices” is also used for exponents.
Positive Integral Exponents
A positive integral exponent is a positive whole-number power such as 2, 3, 4, 5, and so on. When the exponent is positive, the base is multiplied repeatedly.
Example: 24 = 2 × 2 × 2 × 2 = 16.
Example: 92 = 9 × 9 = 81.
Example: 35 = 3 × 3 × 3 × 3 × 3 = 243.
Law 1: Product Rule of Exponents
When two powers have the same base and they are multiplied, keep the same base and add the exponents.
am × an = am+n
Example: 23 × 24 = 23+4 = 27 = 128.
Why this works: 23 means 2 × 2 × 2 and 24 means 2 × 2 × 2 × 2. Altogether, there are seven 2s in multiplication. So the answer is 27.
Another example: x5 × x2 = x7.
Law 2: Quotient Rule of Exponents
When two powers have the same base and they are divided, keep the same base and subtract the exponents.
am ÷ an = am-n, where a is not equal to 0.
Example: 56 ÷ 52 = 56-2 = 54 = 625.
Example: x9 ÷ x4 = x5.
This rule is valid only when the base is the same. We cannot directly subtract powers in 25 ÷ 32 because the bases are different.
Law 3: Power of a Power Rule
When a power is raised to another power, multiply the exponents.
(am)n = amn
Example: (32)4 = 32×4 = 38.
Example: (x5)3 = x15.
Do not add the exponents in this rule. Product rule uses addition, but power of a power uses multiplication.
Law 4: Power of a Product Rule
When a product is raised to a power, the power applies to each factor.
(ab)n = anbn
Example: (2 × 3)3 = 23 × 33 = 8 × 27 = 216.
Check directly: (2 × 3)3 = 63 = 216.
Example: (xy)4 = x4y4.
Law 5: Power of a Quotient Rule
When a fraction is raised to a power, the power applies to both numerator and denominator.
(a/b)n = an/bn, where b is not equal to 0.
Example: (2/3)4 = 24/34 = 16/81.
Example: (5/2)3 = 53/23 = 125/8.
Law 6: Zero Exponent Rule
Any non-zero number raised to the power 0 is equal to 1.
a0 = 1, where a is not equal to 0.
Examples:
- 70 = 1
- 1000 = 1
- (-5)0 = 1
- x0 = 1, if x is not equal to 0
Why does this happen? Look at the pattern: 24 = 16, 23 = 8, 22 = 4, 21 = 2. Each time the exponent decreases by 1, the value is divided by 2. So 20 = 2 ÷ 2 = 1.
Law 7: Negative Exponent Rule
A negative exponent means reciprocal. It does not mean the final answer is always negative.
a-n = 1/an, where a is not equal to 0.
Example: 2-3 = 1/23 = 1/8.
Example: 5-2 = 1/52 = 1/25.
For fractions, the reciprocal is taken.
(a/b)-n = (b/a)n, where a and b are not equal to 0.
Example: (2/3)-4 = (3/2)4 = 81/16.
Example: (5/7)-2 = (7/5)2 = 49/25.
Law 8: Fractional Exponents
A fractional exponent connects powers with roots. The denominator of the fraction shows the root.
a1/2 = square root of a.
a1/3 = cube root of a.
am/n = nth root of am.
Example: 251/2 = square root of 25 = 5.
Example: 271/3 = cube root of 27 = 3.
Example: 163/4 can be solved as fourth root of 16, then cube the result. Fourth root of 16 is 2, and 23 = 8. So 163/4 = 8.
Exponent Rules Summary Table
| Rule | Formula | Example |
|---|---|---|
| Product rule | am × an = am+n | 23 × 22 = 25 |
| Quotient rule | am ÷ an = am-n | 56 ÷ 53 = 53 |
| Power of power | (am)n = amn | (32)4 = 38 |
| Power of product | (ab)n = anbn | (2x)3 = 8x3 |
| Power of quotient | (a/b)n = an/bn | (2/5)2 = 4/25 |
| Zero exponent | a0 = 1 | 90 = 1 |
| Negative exponent | a-n = 1/an | 3-2 = 1/9 |
Exponents with Negative Bases
When the base is negative, brackets are very important.
(-2)4 = (-2) × (-2) × (-2) × (-2) = 16.
(-2)3 = (-2) × (-2) × (-2) = -8.
If the negative base is inside brackets, the exponent applies to the whole negative number. If there are no brackets, the exponent may apply only to the number and not to the minus sign.
Example: (-3)2 = 9, but -32 = -9.
This is a very common mistake in exams. Always notice whether brackets are present.
Powers of 10
Powers of 10 are very useful because our number system is based on 10.
| Power | Value |
|---|---|
| 101 | 10 |
| 102 | 100 |
| 103 | 1000 |
| 104 | 10000 |
| 105 | 100000 |
| 106 | 1000000 |
For positive powers of 10, the exponent tells how many zeros come after 1. For example, 105 = 100000.
For negative powers of 10, the value becomes a decimal fraction.
10-1 = 1/10 = 0.1.
10-2 = 1/100 = 0.01.
10-3 = 1/1000 = 0.001.
Scientific Notation
Scientific notation is used to write very large or very small numbers in a compact form. A number in scientific notation is written as:
a × 10n, where 1 ≤ a < 10 and n is an integer.
Example: 400000 = 4 × 105.
Example: 7500000 = 7.5 × 106.
Example: 0.008 = 8 × 10-3.
Example: 0.00045 = 4.5 × 10-4.
Use positive powers of 10 for large numbers and negative powers of 10 for small decimal numbers.
Squares and Cubes as Exponents
When a number is raised to the power 2, it is called the square of the number.
Examples:
- 12 = 1
- 22 = 4
- 32 = 9
- 42 = 16
- 52 = 25
- 102 = 100
Numbers such as 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are called perfect squares because they are squares of natural numbers.
When a number is raised to the power 3, it is called the cube of the number.
Examples:
- 13 = 1
- 23 = 8
- 33 = 27
- 43 = 64
- 53 = 125
Numbers such as 1, 8, 27, 64, 125, 216, 343, 512, and 729 are called perfect cubes.
Roots Written as Exponents
Roots can also be written using fractional exponents. This is why exponents are important in algebra.
Square root of x = x1/2.
Cube root of x = x1/3.
Fourth root of x = x1/4.
Example: square root of 81 = 811/2 = 9.
Example: cube root of 64 = 641/3 = 4.
Example: 321/5 = 2 because 25 = 32.
How to Simplify Exponential Expressions
To simplify an expression with exponents, first check whether the bases are the same. If the bases are the same, use product rule or quotient rule. If a power is raised to another power, multiply the exponents. If the exponent is negative, convert it into a reciprocal. If the exponent is fractional, convert it into a root.
Example: Simplify x4 × x7.
Same base x, so add exponents.
x4 × x7 = x11.
Example: Simplify y9 ÷ y3.
Same base y, so subtract exponents.
y9 ÷ y3 = y6.
Example: Simplify (a2b3)4.
Apply the power to each factor.
(a2)4(b3)4 = a8b12.
Worked Examples
Example 1
Simplify 43.
43 = 4 × 4 × 4 = 64.
Example 2
Simplify 25 × 23.
Use product rule.
25 × 23 = 28 = 256.
Example 3
Simplify 107 ÷ 104.
Use quotient rule.
107 ÷ 104 = 103 = 1000.
Example 4
Simplify (52)3.
Use power of a power rule.
(52)3 = 56 = 15625.
Example 5
Simplify (2/3)3.
(2/3)3 = 23/33 = 8/27.
Example 6
Simplify 60.
Any non-zero number raised to 0 is 1.
60 = 1.
Example 7
Simplify 4-2.
4-2 = 1/42 = 1/16.
Example 8
Simplify 491/2.
491/2 means square root of 49.
491/2 = 7.
Example 9
Write 900000 in scientific notation.
Move the decimal point after 9. There are 5 places moved.
900000 = 9 × 105.
Example 10
Write 0.00072 in scientific notation.
Move the decimal point to make 7.2. The decimal moved 4 places to the right, so the exponent is negative.
0.00072 = 7.2 × 10-4.
Common Mistakes in Exponents
Mistake 1: Adding exponents when bases are different.
Wrong: 23 × 32 = 65.
Correct: Calculate separately or simplify only if possible. 23 × 32 = 8 × 9 = 72.
Mistake 2: Thinking a negative exponent gives a negative answer.
Wrong: 2-3 = -8.
Correct: 2-3 = 1/8.
Mistake 3: Forgetting brackets with negative bases.
(-4)2 = 16, but -42 = -16.
Mistake 4: Multiplying instead of adding in the product rule.
Wrong: x3 × x4 = x12.
Correct: x3 × x4 = x7.
Mistake 5: Adding exponents in power of a power.
Wrong: (x3)4 = x7.
Correct: (x3)4 = x12.
Exam Style Questions
Question 1
Simplify 34 × 32.
Answer: 36 = 729.
Question 2
Simplify 78 ÷ 75.
Answer: 73 = 343.
Question 3
Simplify (23)2.
Answer: 26 = 64.
Question 4
Simplify (3x)2.
Answer: 9x2.
Question 5
Simplify (a2b3)2.
Answer: a4b6.
Question 6
Simplify 8-1.
Answer: 1/8.
Question 7
Simplify 1251/3.
Answer: 5.
Question 8
Write 0.00006 as a power of 10.
Answer: 6 × 10-5.
Practice Questions
- Find the value of 27.
- Find the value of 63.
- Simplify x5 × x6.
- Simplify a9 ÷ a4.
- Simplify (m3)5.
- Simplify (2x)4.
- Simplify (3/5)2.
- Simplify 110.
- Simplify 10-4.
- Simplify 641/2.
- Simplify 2161/3.
- Write 3500000 in scientific notation.
- Write 0.0009 in scientific notation.
- Simplify y2 × y3 × y4.
- Simplify p10 ÷ p6.
Answers to Practice Questions
- 27 = 128.
- 63 = 216.
- x5 × x6 = x11.
- a9 ÷ a4 = a5.
- (m3)5 = m15.
- (2x)4 = 16x4.
- (3/5)2 = 9/25.
- 110 = 1.
- 10-4 = 1/10000 = 0.0001.
- 641/2 = 8.
- 2161/3 = 6.
- 3500000 = 3.5 × 106.
- 0.0009 = 9 × 10-4.
- y2 × y3 × y4 = y9.
- p10 ÷ p6 = p4.
Frequently Asked Questions
What is an exponent in mathematics?
An exponent is a small number written above and to the right of a base. It tells how many times the base is multiplied by itself.
What is the difference between base and exponent?
The base is the number being multiplied. The exponent tells how many times the base is used in multiplication. In 64, 6 is the base and 4 is the exponent.
What is another name for exponent?
An exponent is also called a power or index. In some books, exponents are studied under the topic “indices”.
What is the value of any non-zero number raised to power 0?
Any non-zero number raised to the power 0 is 1. For example, 120 = 1.
What does a negative exponent mean?
A negative exponent means reciprocal. For example, 3-2 = 1/32 = 1/9.
What does a fractional exponent mean?
A fractional exponent represents a root. For example, 251/2 means square root of 25, which is 5.
When do we add exponents?
We add exponents when powers with the same base are multiplied. For example, a3 × a5 = a8.
When do we subtract exponents?
We subtract exponents when powers with the same base are divided. For example, a7 ÷ a2 = a5.
When do we multiply exponents?
We multiply exponents when a power is raised to another power. For example, (a3)4 = a12.
Why are exponents useful?
Exponents make repeated multiplication shorter. They also help in algebra, square roots, cube roots, scientific notation, compound interest, physics, computer science, and many higher mathematics topics.
More Daily Life Examples of Exponents
Exponents are not only used in textbook questions. They appear in many daily calculations also. If a square garden has side 12 m, then its area is 122 = 144 square metres. If a cube has side 5 cm, then its volume is 53 = 125 cubic centimetres. This is why square and cube formulas use exponents.
In computer memory, powers of 2 are very common. For example, 210 = 1024. That is why 1024 bytes make 1 kilobyte in the binary system. Similarly, 220 is used for about one million bytes. Learning powers of 2 helps in computer science and digital electronics.
In science, exponents help us write large distances and tiny measurements. The speed of light is about 3 × 108 metres per second. The size of very small particles is often written using negative powers of 10. Without exponents, these numbers become long and difficult to read.
In banking and finance, repeated growth can be written using powers. Compound interest uses exponents because the amount grows again and again after every period. Population growth, bacteria growth, and depreciation of value can also be explained with exponents.
Quick Revision Before Solving Questions
Before solving any exponent question, check four things. First, see whether the bases are the same. Second, notice whether the operation is multiplication, division, or power over power. Third, check whether the exponent is zero, negative, or fractional. Fourth, look carefully for brackets, especially when the base is negative.
If you remember these points, most exponent questions become direct. Do not rush to apply a formula. Read the expression slowly, identify the rule, and then simplify step by step.
Conclusion
Exponents are an easy and powerful way to write repeated multiplication. Once you understand the base, exponent, and basic laws, you can simplify many long expressions quickly.
Remember the main ideas: multiply same bases by adding powers, divide same bases by subtracting powers, raise a power to a power by multiplying powers, use reciprocal for negative exponents, and use roots for fractional exponents.