Exponents and Powers - Class 7 Mathematics (CBSE)
Based on the 2025-26 NCERT syllabus for Class 7 Mathematics. This chapter introduces exponential notation and the laws of exponents, enabling students to work with very large and very small numbers efficiently.
1. Why this chapter matters
Exponents are a shorthand for repeated multiplication. They are used in science to express astronomical distances, microscopic sizes, and population figures. In CBSE exams, this chapter contributes 6-8 marks and is critical for Class 8 Exponents and Powers and Class 10 Scientific Notation.
2. What is an exponent?
An exponent tells how many times a base number is multiplied by itself.
In a raised to n:
- a is the base.
- n is the exponent (or power).
- a raised to n means a x a x a x ... (n times).
Example: 2 raised to 5 = 2 x 2 x 2 x 2 x 2 = 32.
Reading exponents
- 2-squared: 2 raised to power 2
- 5-cubed: 5 raised to power 3
- 10 raised to 4: 10 to the power 4
3. Laws of exponents
For non-zero integers a and b, and positive integers m and n:
Law 1: Product of powers
a raised to m x a raised to n = a raised to (m + n)
3-squared x 3-cubed = 3 raised to (2+3) = 3 raised to 5 = 243.
Law 2: Quotient of powers
a raised to m / a raised to n = a raised to (m - n), where m > n.
5 raised to 6 / 5-squared = 5 raised to (6-2) = 5 raised to 4 = 625.
Law 3: Power of a power
(a raised to m) raised to n = a raised to (m x n).
(2-squared)-cubed = 2 raised to (2 x 3) = 2 raised to 6 = 64.
Law 4: Power of a product
(a x b) raised to n = a raised to n x b raised to n.
(2 x 3)-cubed = 2-cubed x 3-cubed = 8 x 27 = 216.
Law 5: Power of a quotient
(a/b) raised to n = a raised to n / b raised to n.
(6/2)-cubed = 6-cubed / 2-cubed = 216/8 = 27.
4. Zero exponent
Any non-zero number raised to power 0 is 1.
a raised to 0 = 1, where a is not zero.
5 raised to 0 = 1, (-3) raised to 0 = 1, (2/7) raised to 0 = 1.
5. Negative exponent
a raised to (-n) = 1 / (a raised to n), where a is not zero.
2 raised to -3 = 1 / 2-cubed = 1/8. (-3) raised to -2 = 1 / (-3)-squared = 1/9.
6. Standard form (scientific notation)
Very large or very small numbers are expressed in standard form as:
A x 10 raised to n, where 1 is less than or equal to A is less than 10, and n is an integer.
Examples
- 5,00,000 = 5 x 10 raised to 5
- 3,18,00,00,000 = 3.18 x 10 raised to 9
- 0.0007 = 7 x 10 raised to -4
Converting to standard form
For numbers greater than 1: Count how many places the decimal moves left. For numbers less than 1: Count how many places the decimal moves right (negative exponent).
Large numbers in standard form
| Large number | Standard form |
|---|---|
| Distance from Earth to Sun: 149,600,000 km | 1.496 x 10 raised to 8 km |
| Speed of light: 300,000,000 m/s | 3 x 10 raised to 8 m/s |
| Mass of Earth: 5,970,000,000,000,000,000,000,000 kg | 5.97 x 10 raised to 24 kg |
7. Expanded form using exponents
Writing a number as the sum of powers of 10.
735 = 7 x 10-squared + 3 x 10 + 5 x 10 raised to 0.
8. Laws of exponents summary table
| Law | Formula | Example |
|---|---|---|
| Product | a power m x a power n = a power (m+n) | 2-squared x 2-cubed = 2 raised to 5 |
| Quotient | a power m / a power n = a power (m-n) | 5 raised to 4 / 5-squared = 5-squared |
| Power of power | (a power m) power n = a power (m x n) | (3-squared)-cubed = 3 raised to 6 |
| Product power | (ab) power n = a power n x b power n | (2 x 5)-cubed = 2-cubed x 5-cubed |
| Quotient power | (a/b) power n = a power n / b power n | (4/2)-cubed = 4-cubed / 2-cubed |
| Zero exponent | a power 0 = 1 | 7 power 0 = 1 |
| Negative exponent | a power -n = 1/(a power n) | 2 power -3 = 1/8 |
9. Worked examples
Example 1: Simplify 2-cubed x 2-squared / 2 raised to 4.
2-cubed x 2-squared = 2 raised to (3+2) = 2 raised to 5. 2 raised to 5 / 2 raised to 4 = 2 raised to (5-4) = 2 raised to 1 = 2.
Example 2: Simplify (2-cubed x 3-squared)-squared.
2-cubed = 8, 3-squared = 9. 8 x 9 = 72. 72-squared = 5184. Or using law: (2-cubed)-squared x (3-squared)-squared = 2 raised to 6 x 3 raised to 4 = 64 x 81 = 5184.
Example 3: Express 0.00000032 in standard form.
0.00000032 = 3.2 / 10,000,000 = 3.2 x 10 raised to -7.
Example 4: Evaluate 2 power -3 x 2 power -4 / 2 power -5.
2 power (-3-4-(-5)) = 2 power (-3-4+5) = 2 power (-2) = 1/2-squared = 1/4.
10. Common mistakes and how to fix them
| Mistake | Fix |
|---|---|
| Adding exponents when multiplying different bases | You can only add exponents when BASES are same |
| Thinking a power 0 = 0 | Any non-zero number to power 0 equals 1 |
| Using negative exponent as negative number | a power -n = 1/(a power n), not -a power n |
| Writing (3x) power 2 as 3x-squared | (3x)-squared = 9x-squared, not 3x-squared |
| Missing the 'between 1 and 10' rule in standard form | A must satisfy 1 is less than or equal to A is less than 10 |
11. CBSE exam focus
| Question type | Marks | Frequency |
|---|---|---|
| Simplify using laws of exponents | 2-3 marks | 1-2 questions |
| Express in standard form | 2 marks | 1 question |
| Evaluate expressions with exponents | 3 marks | 1 question |
| Compare large numbers in standard form | 2 marks | Occasional |
| Expanded form using powers of 10 | 2 marks | 1 question |
12. Self-test
- Simplify: 3 raised to 5 x 3 raised to 7 / 3 raised to 10.
- Simplify: (2-cubed x 5-squared)-squared.
- Express 0.000000459 in standard form.
- Evaluate: (-2) power -3.
- Express the distance 150,000,000 km in standard form.
- Which is greater: 2 raised to 5 or 5-squared? Verify.
13. Answer key
- 3 raised to (5+7-10) = 3-squared = 9.
- (2-cubed)-squared x (5-squared)-squared = 2 raised to 6 x 5 raised to 4 = 64 x 625 = 40000.
- 4.59 x 10 raised to -7.
- (-2) power -3 = 1 / (-2)-cubed = 1 / (-8) = -1/8.
- 1.5 x 10 raised to 8 km.
- 2 raised to 5 = 32. 5-squared = 25. So 2 raised to 5 is greater.
14. Quick revision
- a power n means a multiplied by itself n times.
- Product: add exponents for same base.
- Quotient: subtract exponents for same base.
- (a power m) power n = a power (m x n).
- Any non-zero base to power 0 = 1.
- a power -n = 1 / (a power n).
- Standard form: A x 10 power n, where 1 is less than or equal to A is less than 10.
