Power rule

Key Notes :

🧮 Power Rule

🔹 What is the Power Rule?

The Power Rule is a rule used to simplify and solve problems involving exponents (also called powers).

📘 Basic Power Rules

1. Product of Powers Rule

a^m×aⁿ=a^m+n

➤ When you multiply powers with the same base, add the exponents.

• Example: 3²×3⁴=3²⁺⁴=3⁶

2. Quotient of Powers Rule

a^m/aⁿ=a^m−n

➤ When you divide powers with the same base, subtract the exponents.

• Example: 5⁶÷5²=5⁶⁻²=5⁴

3. Power of a Power Rule

(a^m)ⁿ=a^m×n

➤ When you have a power raised to another power, multiply the exponents.

• Example: (2³)²=2^3×2=2⁶

4. Power of a Product Rule

(ab)ⁿ=aⁿ×bⁿ

➤ When a product is raised to a power, apply the power to each factor.

• Example: (2×3)⁴=2⁴×3⁴

5. Power of a Quotient Rule

(a/b)ⁿ=aⁿ/bⁿ

➤ When a fraction is raised to a power, apply the power to both numerator and denominator.

• Example: (2/3)³=2³/3³=8/27

6. Zero Exponent Rule

a⁰=1(a≠0)

➤ Any non-zero number raised to the power of zero equals 1.

• Example: 9⁰=1

7. Negative Exponent Rule

a⁻ⁿ=1/aⁿ

​➤ A negative exponent means take the reciprocal of the base.

• Example: 2⁻³=1/2³=1/8

⚡ Important Points to Remember

• The base must be the same to apply most rules.
• Exponents show repeated multiplication.
• Always simplify step-by-step using these rules.

🧠 Example Problems

1. (x³)⁴=x^3×4=x¹²
2. a⁵×a²=a⁵⁺²=a⁷
3. y⁸/y³=y⁸⁻³=y⁵
4. (2x)³=2³.x³=8x³
5. (3²)⁰=1

🎯 Summary Table

Rule Name	Formula	Example
Product of Powers	am×an=am+n	x2.x3=x5
Quotient of Powers	am/an=am−n	y6÷y2=y4
Power of a Power	(am)n=amn	(32)3=36
Power of a Product	(ab)n=anbn	(2x)3=8x3
Power of a Quotient	(a/b)n=an/bn​	(x/2)2=x2/4
Zero Exponent	a0=1	50=1
Negative Exponent	a−n=1/an	4−2=1/16

Learn with an example