Introduction to Exponents
When you multiply a number by itself several times, writing it out becomes long and difficult. For example, 2 × 2 × 2 × 2 × 2 is the same as 2 multiplied by itself 5 times. Instead of writing this long multiplication, we use a short form called exponents.
We write 2 × 2 × 2 × 2 × 2 as 25. Here, 2 is the number being multiplied and 5 tells us how many times it is multiplied. This saves space and makes calculations easier to read.
In Class 7 Mathematics (NCERT), exponents are studied in the chapter Exponents and Powers. You will learn what base and exponent mean, how to read and write powers, and how exponents help express very large numbers in a simple way.
What is Introduction to Exponents?
Definition: An exponent tells us how many times a number (called the base) is multiplied by itself.
The expression is written as:
an
Where:
- a = the base (the number being multiplied)
- n = the exponent or power or index (how many times the base is multiplied by itself)
How to read:
- an is read as "a raised to the power n" or "a to the power n"
- 25 is read as "2 raised to the power 5" or "2 to the power 5"
- 34 is read as "3 raised to the power 4"
Special names:
- a2 is called "a squared" (because the area of a square with side a = a2)
- a3 is called "a cubed" (because the volume of a cube with side a = a3)
Important:
- an does NOT mean a × n. It means a multiplied by itself n times.
- 25 = 2 × 2 × 2 × 2 × 2 = 32 (NOT 2 × 5 = 10)
- The base can be any number — whole number, fraction, or decimal.
Introduction to Exponents Formula
Exponential Form:
an = a × a × a × ... (n times)
Where:
- a = base
- n = exponent (a positive whole number)
Common powers to remember:
- Any number raised to the power 1 is the number itself: a1 = a
- Any number raised to the power 0 is 1: a0 = 1 (where a is not 0)
- 1 raised to any power is always 1: 1n = 1
Powers of small numbers:
- 21 = 2, 22 = 4, 23 = 8, 24 = 16, 25 = 32, 26 = 64, 27 = 128, 28 = 256, 29 = 512, 210 = 1024
- 31 = 3, 32 = 9, 33 = 27, 34 = 81, 35 = 243
- 51 = 5, 52 = 25, 53 = 125, 54 = 625
- 101 = 10, 102 = 100, 103 = 1000, 104 = 10000
Derivation and Proof
Understanding why exponents work:
From repeated addition to repeated multiplication:
- Multiplication is repeated addition: 3 × 4 = 3 + 3 + 3 + 3 (add 3 four times).
- In the same way, an exponent is repeated multiplication: 34 = 3 × 3 × 3 × 3 (multiply 3 four times).
- Just as multiplication gives a short form for addition, exponents give a short form for multiplication.
Why a0 = 1:
- 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 (the base).
- So 20 = 21 ÷ 2 = 2 ÷ 2 = 1.
- This works for any base: 50 = 1, 100 = 1, 1000 = 1.
Why a1 = a:
- a1 means "multiply a by itself 1 time" — that is just a itself.
- So 71 = 7, 1001 = 100.
Types and Properties
Types of exponent expressions:
1. Positive base, positive exponent:
- Example: 34 = 3 × 3 × 3 × 3 = 81
- The result is always positive.
2. Negative base, positive exponent:
- Example: (−2)3 = (−2) × (−2) × (−2) = −8
- If the exponent is odd, the result is negative.
- Example: (−2)4 = (−2) × (−2) × (−2) × (−2) = 16
- If the exponent is even, the result is positive.
3. Fractional base:
- Example: (2/3)3 = (2/3) × (2/3) × (2/3) = 8/27
- Raise both numerator and denominator to the power separately.
4. Exponent 0:
- Any non-zero number raised to the power 0 equals 1.
- 50 = 1, (−3)0 = 1, (7/8)0 = 1
5. Exponent 1:
- Any number raised to the power 1 is the number itself.
- 91 = 9, (−4)1 = −4
Important note:
- (−2)4 and −24 are NOT the same.
- (−2)4 = (−2) × (−2) × (−2) × (−2) = +16
- −24 = −(24) = −(16) = −16
- The brackets make a big difference!
Solved Examples
Example 1: Example 1: Writing in exponential form
Problem: Write 5 × 5 × 5 × 5 × 5 × 5 in exponential form.
Solution:
Given:
- The number 5 is multiplied by itself 6 times.
In exponential form:
- Base = 5
- Exponent = 6
- 5 × 5 × 5 × 5 × 5 × 5 = 56
Answer: 5 × 5 × 5 × 5 × 5 × 5 = 56
Example 2: Example 2: Finding the value of a power
Problem: Find the value of 35.
Solution:
Given:
- Base = 3, Exponent = 5
Calculate step by step:
- 35 = 3 × 3 × 3 × 3 × 3
- = 9 × 3 × 3 × 3
- = 27 × 3 × 3
- = 81 × 3
- = 243
Answer: 35 = 243
Example 3: Example 3: Identifying base and exponent
Problem: In each expression, identify the base and the exponent: (a) 74, (b) 123, (c) (−5)6, (d) (1/2)8
Solution:
- (a) 74: Base = 7, Exponent = 4. Read as: "7 to the power 4."
- (b) 123: Base = 12, Exponent = 3. Read as: "12 cubed."
- (c) (−5)6: Base = −5, Exponent = 6. The negative sign is part of the base because of the brackets.
- (d) (1/2)8: Base = 1/2, Exponent = 8. The fraction 1/2 is the base.
Answer: The base is the number being repeated and the exponent is how many times.
Example 4: Example 4: Expanded form using powers of 10
Problem: Write 47,305 in expanded form using powers of 10.
Solution:
Given:
- 47,305 = 4 ten-thousands + 7 thousands + 3 hundreds + 0 tens + 5 ones
Using powers of 10:
- = 4 × 10,000 + 7 × 1,000 + 3 × 100 + 0 × 10 + 5 × 1
- = 4 × 104 + 7 × 103 + 3 × 102 + 0 × 101 + 5 × 100
Answer: 47,305 = 4 × 104 + 7 × 103 + 3 × 102 + 0 × 101 + 5 × 100
Example 5: Example 5: Comparing exponent expressions
Problem: Which is greater: 28 or 82?
Solution:
Calculate 28:
- 28 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256
Calculate 82:
- 82 = 8 × 8 = 64
Compare:
- 256 > 64
- So 28 > 82
Answer: 28 is greater. (Note: 28 = 256 and 82 = 64.)
Example 6: Example 6: Negative base with odd and even exponents
Problem: Find the value of (a) (−3)4 and (b) (−3)5.
Solution:
(a) (−3)4:
- = (−3) × (−3) × (−3) × (−3)
- = (+9) × (−3) × (−3)
- = (−27) × (−3)
- = +81
(b) (−3)5:
- = (−3) × (−3) × (−3) × (−3) × (−3)
- = (+81) × (−3)
- = −243
Answer: (−3)4 = 81 (positive, even exponent). (−3)5 = −243 (negative, odd exponent).
Example 7: Example 7: Expressing a number as a power of a prime
Problem: Express 125 as a power of 5.
Solution:
Find how many times 5 multiplies to give 125:
- 5 × 5 = 25
- 25 × 5 = 125
- So 125 = 5 × 5 × 5 = 53
Verify: 53 = 125 ✓
Answer: 125 = 53
Example 8: Example 8: Prime factorisation using exponents
Problem: Write the prime factorisation of 360 using exponents.
Solution:
Step 1: Find prime factors:
- 360 ÷ 2 = 180
- 180 ÷ 2 = 90
- 90 ÷ 2 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
Step 2: Write using exponents:
- 360 = 2 × 2 × 2 × 3 × 3 × 5
- 360 = 23 × 32 × 51
Answer: 360 = 23 × 32 × 5
Example 9: Example 9: Fractional base
Problem: Find the value of (2/3)4.
Solution:
Given:
- Base = 2/3, Exponent = 4
Calculate:
- (2/3)4 = (2/3) × (2/3) × (2/3) × (2/3)
- Numerator: 2 × 2 × 2 × 2 = 24 = 16
- Denominator: 3 × 3 × 3 × 3 = 34 = 81
- (2/3)4 = 16/81
Answer: (2/3)4 = 16/81
Example 10: Example 10: Expressing large numbers using exponents
Problem: The distance from the Earth to the Sun is about 150,000,000 km. Express this using powers of 10.
Solution:
Given:
- 150,000,000 km
Write in terms of powers of 10:
- 150,000,000 = 15 × 10,000,000
- = 15 × 107
- = 1.5 × 108
Answer: The Earth-Sun distance is about 1.5 × 108 km.
Real-World Applications
Real-world uses of exponents:
- Large numbers in science: The speed of light is about 3 × 108 m/s. Without exponents, you would have to write 300,000,000 m/s every time.
- Computer memory: Computer storage is measured in powers of 2. 1 KB = 210 bytes = 1024 bytes. 1 MB = 220 bytes. 1 GB = 230 bytes.
- Population and geography: India's population is about 1.4 × 109. The Earth's age is about 4.5 × 109 years.
- Microbiology: Bacteria can double every 20 minutes. After n doublings, 1 bacterium becomes 2n bacteria. After 10 hours (30 doublings), there are 230 = over 1 billion bacteria!
- Area and volume: The area of a square with side s is s2. The volume of a cube with side s is s3. That is why we say "squared" and "cubed."
- Prime factorisation: Exponents make it easy to write prime factorisations: 7200 = 25 × 32 × 52.
- Money and interest: Compound interest uses exponents. Amount = P(1 + r/100)n.
Key Points to Remember
- An exponent tells how many times a base is multiplied by itself: an = a × a × ... (n times).
- Base = the number being multiplied. Exponent = how many times.
- a2 is called "a squared." a3 is called "a cubed."
- a1 = a (any number to the power 1 is itself).
- a0 = 1 (any non-zero number to the power 0 is 1).
- 1n = 1 (1 raised to any power is always 1).
- Negative base with even exponent gives a positive result.
- Negative base with odd exponent gives a negative result.
- (−a)n and −an are NOT the same. Brackets matter!
- Exponents help express very large and very small numbers using powers of 10.
Practice Problems
- Write 7 × 7 × 7 × 7 in exponential form.
- Find the value of 44.
- Express 512 as a power of 2.
- Which is greater: 35 or 53? Find both values.
- Find the value of (−2)6 and (−2)7.
- Write the prime factorisation of 1080 using exponents.
- Find the value of (3/4)3.
- Express 9,00,00,000 using powers of 10.
Frequently Asked Questions
Q1. What is an exponent?
An exponent tells how many times a number (the base) is multiplied by itself. In 2<sup>5</sup>, the base is 2 and the exponent is 5, meaning 2 × 2 × 2 × 2 × 2 = 32.
Q2. What is the difference between 2<sup>5</sup> and 2 × 5?
2<sup>5</sup> means 2 multiplied by itself 5 times = 2 × 2 × 2 × 2 × 2 = 32. But 2 × 5 means 2 added 5 times = 10. They are very different operations.
Q3. Why is any number to the power 0 equal to 1?
Look at the pattern: 2<sup>3</sup> = 8, 2<sup>2</sup> = 4, 2<sup>1</sup> = 2. Each time the exponent decreases by 1, we divide by the base (2). So 2<sup>0</sup> = 2<sup>1</sup> ÷ 2 = 2 ÷ 2 = 1. This works for any non-zero base.
Q4. What happens when a negative number is raised to a power?
If the exponent is even, the result is positive: (−3)<sup>2</sup> = 9. If the exponent is odd, the result is negative: (−3)<sup>3</sup> = −27. This is because multiplying two negatives gives a positive.
Q5. Is (−5)<sup>2</sup> the same as −5<sup>2</sup>?
No. (−5)<sup>2</sup> = (−5) × (−5) = +25. But −5<sup>2</sup> = −(5<sup>2</sup>) = −(25) = −25. The brackets make the negative sign part of the base.
Q6. What is 1 raised to any power?
1 raised to any power is always 1. Because 1 × 1 × 1 × ... = 1 no matter how many times you multiply. So 1<sup>5</sup> = 1, 1<sup>100</sup> = 1, 1<sup>1000</sup> = 1.
Q7. How are exponents used to write large numbers?
Large numbers can be written using powers of 10. For example, 5,00,00,000 = 5 × 10<sup>7</sup>. The speed of light is 3 × 10<sup>8</sup> m/s. This makes large numbers easier to read and compare.
Q8. Can fractions be raised to a power?
Yes. (a/b)<sup>n</sup> = a<sup>n</sup>/b<sup>n</sup>. For example, (2/3)<sup>3</sup> = 2<sup>3</sup>/3<sup>3</sup> = 8/27. Raise the numerator and denominator to the power separately.
Q9. What is the square of a number?
The square of a number is the number multiplied by itself once: a<sup>2</sup> = a × a. For example, 5<sup>2</sup> = 5 × 5 = 25. It is called 'squared' because it gives the area of a square with that side length.
Q10. What is the cube of a number?
The cube of a number is the number multiplied by itself three times: a<sup>3</sup> = a × a × a. For example, 4<sup>3</sup> = 4 × 4 × 4 = 64. It is called 'cubed' because it gives the volume of a cube with that side length.
