Laws of Exponents
In the previous topic, you learned that exponents are a short way to write repeated multiplication. Now, when you multiply, divide, or raise powers of numbers written in exponential form, there are simple rules that make the calculation much easier.
These rules are called the Laws of Exponents (also known as exponent rules or index laws). Instead of expanding every power and multiplying each time, these laws let you work directly with the exponents.
For example, to find 23 × 24, you do not need to calculate 8 × 16. You can simply add the exponents: 23+4 = 27 = 128. These laws save a lot of time, especially with large numbers.
There are seven main laws of exponents. In this topic, you will learn each law, understand why it works, and practise using them to simplify expressions. Mastering these laws is important because they are used everywhere — in science, computing, and higher mathematics.
What is Laws of Exponents?
Definition: The laws of exponents are a set of rules that tell us how to simplify expressions involving powers of the same base.
These laws apply when:
- The bases are the same (for product and quotient rules)
- The base is not zero (for the zero exponent rule)
- The exponents are whole numbers (in Class 7)
Laws of Exponents Formula
The Seven Laws of Exponents:
Law 1: Product of Powers (same base)
am × an = am+n
When multiplying powers with the same base, add the exponents.
Law 2: Quotient of Powers (same base)
am ÷ an = am−n (where m > n)
When dividing powers with the same base, subtract the exponents.
Law 3: Power of a Power
(am)n = am×n
When raising a power to another power, multiply the exponents.
Law 4: Power of a Product
(a × b)n = an × bn
The power of a product equals the product of the powers.
Law 5: Power of a Quotient
(a/b)n = an/bn
The power of a quotient equals the quotient of the powers.
Law 6: Zero Exponent
a0 = 1 (where a ≠ 0)
Any non-zero number raised to the power 0 is 1.
Law 7: Exponent 1
a1 = a
Any number raised to the power 1 is the number itself.
Derivation and Proof
Understanding why these laws work:
Why am × an = am+n:
- am = a × a × a × ... (m times)
- an = a × a × a × ... (n times)
- am × an = (a × a × ... m times) × (a × a × ... n times)
- = a × a × a × ... (m + n times)
- = am+n
Why am ÷ an = am−n:
- am ÷ an = am / an
- = (a × a × ... m times) / (a × a × ... n times)
- Cancel n factors of a from the top and bottom.
- You are left with (m − n) factors of a on top.
- = am−n
Why (am)n = amn:
- (am)n = am × am × am × ... (n times)
- Using the product rule: add the exponents m + m + m + ... (n times)
- = am×n
Types and Properties
Common types of problems using laws of exponents:
1. Simplify using product rule:
- 53 × 54 = 53+4 = 57
- Same base, so add exponents.
2. Simplify using quotient rule:
- 79 ÷ 75 = 79−5 = 74
- Same base, so subtract exponents.
3. Simplify power of a power:
- (32)4 = 32×4 = 38
- Multiply the exponents.
4. Simplify power of a product:
- (2 × 5)3 = 23 × 53 = 8 × 125 = 1000
5. Find the value of the exponent:
- If 2x = 64, find x. Since 64 = 26, x = 6.
6. Express using a single exponent:
- Combine multiple laws in one problem.
Solved Examples
Example 1: Example 1: Product of powers
Problem: Simplify 24 × 23.
Solution:
Given:
- Same base (2), exponents 4 and 3.
Using Law 1 (Product Rule):
- am × an = am+n
- 24 × 23 = 24+3 = 27
Value: 27 = 128
Verify: 24 = 16, 23 = 8, 16 × 8 = 128 ✓
Answer: 24 × 23 = 27 = 128
Example 2: Example 2: Quotient of powers
Problem: Simplify 58 ÷ 55.
Solution:
Given:
- Same base (5), exponents 8 and 5.
Using Law 2 (Quotient Rule):
- am ÷ an = am−n
- 58 ÷ 55 = 58−5 = 53
Value: 53 = 125
Answer: 58 ÷ 55 = 53 = 125
Example 3: Example 3: Power of a power
Problem: Simplify (43)2.
Solution:
Using Law 3 (Power of a Power):
- (am)n = am×n
- (43)2 = 43×2 = 46
Value: 46 = 4096
Verify: 43 = 64, then 642 = 64 × 64 = 4096 ✓
Answer: (43)2 = 46 = 4096
Example 4: Example 4: Power of a product
Problem: Simplify (3 × 7)4.
Solution:
Using Law 4 (Power of a Product):
- (a × b)n = an × bn
- (3 × 7)4 = 34 × 74
- = 81 × 2401
- = 194481
Verify: 3 × 7 = 21. Then 214 = 21 × 21 × 21 × 21 = 441 × 441 = 194481 ✓
Answer: (3 × 7)4 = 34 × 74 = 194481
Example 5: Example 5: Power of a quotient
Problem: Simplify (2/5)3.
Solution:
Using Law 5 (Power of a Quotient):
- (a/b)n = an/bn
- (2/5)3 = 23/53
- = 8/125
Answer: (2/5)3 = 8/125
Example 6: Example 6: Using zero exponent rule
Problem: Find the value of 75 ÷ 75 using the quotient rule.
Solution:
Method 1: Direct calculation
- 75 ÷ 75 = any number divided by itself = 1
Method 2: Using quotient rule
- 75 ÷ 75 = 75−5 = 70
Comparing both methods:
- 70 = 1
Answer: 75 ÷ 75 = 70 = 1. This proves that any non-zero number to the power 0 is 1.
Example 7: Example 7: Combining multiple laws
Problem: Simplify: (23 × 25) ÷ 24.
Solution:
Step 1: Simplify the numerator using product rule:
- 23 × 25 = 23+5 = 28
Step 2: Divide using quotient rule:
- 28 ÷ 24 = 28−4 = 24
Value: 24 = 16
Answer: (23 × 25) ÷ 24 = 24 = 16
Example 8: Example 8: Finding the missing exponent
Problem: If 3x × 34 = 310, find x.
Solution:
Using the product rule:
- 3x × 34 = 3x+4
Given that this equals 310:
- 3x+4 = 310
- Since the bases are equal, the exponents must be equal:
- x + 4 = 10
- x = 10 − 4 = 6
Verify: 36 × 34 = 310 ✓
Answer: x = 6
Example 9: Example 9: Simplifying expressions with different laws
Problem: Simplify: (52)3 × 54 ÷ 57.
Solution:
Step 1: Power of a power:
- (52)3 = 52×3 = 56
Step 2: Multiply using product rule:
- 56 × 54 = 56+4 = 510
Step 3: Divide using quotient rule:
- 510 ÷ 57 = 510−7 = 53
Value: 53 = 125
Answer: The simplified expression is 53 = 125
Example 10: Example 10: Expressing as a single power
Problem: Express 83 × 42 × 2 as a power of 2.
Solution:
Step 1: Write each number as a power of 2:
- 8 = 23, so 83 = (23)3 = 29
- 4 = 22, so 42 = (22)2 = 24
- 2 = 21
Step 2: Multiply using product rule:
- 29 × 24 × 21 = 29+4+1 = 214
Answer: 83 × 42 × 2 = 214
Real-World Applications
Real-world uses of laws of exponents:
- Scientific notation: Scientists write very large or small numbers using powers of 10. Laws of exponents help multiply and divide such numbers quickly. For example, (3 × 108) × (2 × 105) = 6 × 1013.
- Computer science: Binary numbers use powers of 2. Understanding exponent laws helps in memory calculations and data operations.
- Finance: Compound interest calculations use the formula A = P(1 + r)n. Laws of exponents help simplify when comparing different time periods.
- Physics: Many physical formulas involve powers. The inverse square law for gravity states force is proportional to 1/r2. Simplifying such expressions uses exponent laws.
- Simplifying large calculations: Instead of multiplying 1015 × 1012 the long way, the product rule gives 1027 instantly.
Key Points to Remember
- Product rule: am × an = am+n (same base, add exponents).
- Quotient rule: am ÷ an = am−n (same base, subtract exponents).
- Power of a power: (am)n = am×n (multiply exponents).
- Power of a product: (a × b)n = an × bn.
- Power of a quotient: (a/b)n = an/bn.
- Zero exponent: a0 = 1 (for a ≠ 0).
- These laws work ONLY when the bases are the same (for product and quotient rules).
- You CANNOT add exponents when the bases are different: 23 × 34 ≠ 67.
- Always simplify step by step: apply one law at a time.
- To express a number as a single power of a prime, first write all factors as powers of that prime.
Practice Problems
- Simplify: 65 × 63.
- Simplify: 97 ÷ 94.
- Simplify: (72)5.
- Simplify: (4 × 3)5.
- Express (3/7)4 without brackets.
- Find x if: 2x ÷ 23 = 25.
- Simplify: (103)2 × 104 ÷ 108.
- Express 272 × 93 as a single power of 3.
Frequently Asked Questions
Q1. What are the laws of exponents?
The laws of exponents are rules for simplifying expressions with powers. The main laws are: product rule (add exponents when multiplying same base), quotient rule (subtract exponents when dividing same base), and power of a power rule (multiply exponents).
Q2. Can you add exponents when the bases are different?
No. The product rule a<sup>m</sup> × a<sup>n</sup> = a<sup>m+n</sup> only works when the bases are the SAME. 2<sup>3</sup> × 3<sup>2</sup> cannot be simplified by adding exponents. You must calculate each separately: 8 × 9 = 72.
Q3. What is the difference between product rule and power of a power rule?
Product rule: a<sup>m</sup> × a<sup>n</sup> = a<sup>m+n</sup> — you ADD exponents when multiplying powers with the same base. Power of a power: (a<sup>m</sup>)<sup>n</sup> = a<sup>mn</sup> — you MULTIPLY exponents when raising a power to another power.
Q4. Why does a<sup>m</sup> ÷ a<sup>n</sup> = a<sup>m−n</sup>?
When you divide, you cancel common factors. a<sup>m</sup>/a<sup>n</sup> has m factors of a on top and n factors of a on the bottom. Cancel n factors from both, leaving (m − n) factors of a on top. So the result is a<sup>m−n</sup>.
Q5. Is 0<sup>0</sup> defined?
In Class 7 mathematics, 0<sup>0</sup> is considered undefined. The zero exponent rule a<sup>0</sup> = 1 requires a ≠ 0. This is a special case that is studied in higher mathematics.
Q6. How do you simplify expressions with multiple laws?
Apply one law at a time, working step by step. First handle power of a power, then product rule, then quotient rule. For example: (2<sup>3</sup>)<sup>2</sup> × 2<sup>4</sup> ÷ 2<sup>5</sup> → 2<sup>6</sup> × 2<sup>4</sup> ÷ 2<sup>5</sup> → 2<sup>10</sup> ÷ 2<sup>5</sup> → 2<sup>5</sup> = 32.
Q7. Can the laws of exponents be used with fractions as bases?
Yes. All laws apply to fractional bases. For example, (2/3)<sup>4</sup> × (2/3)<sup>2</sup> = (2/3)<sup>6</sup> using the product rule. And (2/3)<sup>3</sup> = 2<sup>3</sup>/3<sup>3</sup> = 8/27 using the power of a quotient rule.
Q8. What happens when the exponent is 1?
Any number to the power 1 is the number itself: a<sup>1</sup> = a. This is sometimes called the identity law. For example, 5<sup>1</sup> = 5, (3/4)<sup>1</sup> = 3/4.
