Laws Of Exponent

Unit: Exponents & Powers

Chapter: Laws Of Exponents

Reference: – Review of Exponents, Law 1: Product Rule (a^m × a^n = a^(m+n)), Law 2: Quotient Rule (a^m ÷ a^n = a^(m-n)), Law 3: Power of a Power ((a^m)^n = a^(m×n)), Law 4: Power of a Product ((ab)^m = a^m b^m), Law 5: Power of a Quotient ((a/b)^m = a^m/b^m), Zero Exponent Rule (a⁰ = 1), Negative Exponent Rule (a^(-n) = 1/a^n), Combining Multiple Laws, Solved Examples, Odd-One-Out Problems, Common Mistakes

After studying this chapter, you should be able to understand:

• All Seven Laws of Exponents
• How to Apply Multiple Laws Together
• When to Use Each Law
• Simplify Expressions Using Laws of Exponents

Introduction to Laws of Exponents

Definition

The laws of exponents are rules that tell us how to simplify expressions involving powers (exponents). These rules apply when the bases are the same (for multiplication and division) or when we raise a power to another power, a product to a power, or a quotient to a power.

When we use the laws of exponents, we essentially ask:

"Which rule applies here? How can I simplify this expression?"

Once we master these laws, we can simplify complex exponential expressions quickly and correctly.

Importance of Laws of Exponents

• Essential for algebra, calculus, and higher mathematics
• Used in scientific notation (working with very large/small numbers)
• Helps simplify expressions in physics, chemistry, and engineering
• Foundation for exponential growth and decay problems

Example

Using laws of exponents: 2³ × 2⁴ = 2⁷, 2⁷ ÷ 2³ = 2⁴, (2³)⁴ = 2¹², (2x)³ = 8x³

Subtopics

1. Law 1 – Product Rule (Same Base)

When multiplying two powers with the same base, add the exponents.

Formula: a^m × a^n = a^(m + n)

Example 1: 3⁵ × 3² = 3^(5+2) = 3⁷

Example 2: x⁴ × x⁶ = x^(4+6) = x¹⁰

Example 3: 2 × 2³ = 2¹ × 2³ = 2^(1+3) = 2⁴ = 16

Example 4: 5² × 5³ × 5⁴ = 5^(2+3+4) = 5⁹

2. Law 2 – Quotient Rule (Same Base)

When dividing two powers with the same base, subtract the exponents (numerator exponent minus denominator exponent).

Formula: a^m ÷ a^n = a^(m – n) (a ≠ 0)

Example 1: 7⁶ ÷ 7² = 7^(6-2) = 7⁴

Example 2: x⁹ ÷ x⁴ = x^(9-4) = x⁵

Example 3: 10⁵ ÷ 10³ = 10^(5-3) = 10² = 100

Example 4: 2⁴ ÷ 2⁴ = 2^(4-4) = 2⁰ = 1

3. Law 3 – Power of a Power

When raising a power to another power, multiply the exponents.

Formula: (a^m)^n = a^(m × n)

Example 1: (2³)⁴ = 2^(3×4) = 2¹²

Example 2: (x²)⁵ = x^(2×5) = x¹⁰

Example 3: (5²)³ = 5^(2×3) = 5⁶ = 15625

Example 4: [(3²)³]⁴ = 3^(2×3×4) = 3²⁴

4. Law 4 – Power of a Product

When raising a product to a power, raise each factor to that power.

Formula: (ab)^m = a^m × b^m

Example 1: (2 × 5)³ = 2³ × 5³ = 8 × 125 = 1000 (Check: 10³ = 1000)

Example 2: (3x)⁴ = 3⁴ × x⁴ = 81x⁴

Example 3: (2y)⁵ = 2⁵ × y⁵ = 32y⁵

Example 4: (4ab)³ = 4³ × a³ × b³ = 64a³b³

5. Law 5 – Power of a Quotient

When raising a quotient to a power, raise both numerator and denominator to that power.

Formula: (a/b)^m = a^m / b^m (b ≠ 0)

Example 1: (3/4)² = 3² / 4² = 9/16

Example 2: (x/y)³ = x³ / y³ (y ≠ 0)

Example 3: (2x/5)⁴ = (2x)⁴ / 5⁴ = 16x⁴ / 625

Example 4: (5/2)³ = 5³ / 2³ = 125/8 = 15.625

6. Law 6 – Zero Exponent Rule

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

Formula: a⁰ = 1 (a ≠ 0)

Example 1: 7⁰ = 1

Example 2: 100⁰ = 1

Example 3: x⁰ = 1 (x ≠ 0)

Example 4: (3x)⁰ = 1 (3x ≠ 0)

Example 5: 0⁰ is undefined (not covered in Grade 8)

7. Law 7 – Negative Exponent Rule

A negative exponent means take the reciprocal of the base raised to the positive exponent.

Formula: a^(-n) = 1/a^n (a ≠ 0)

Also, 1/a^(-n) = a^n

Example 1: 2⁻³ = 1/2³ = 1/8

Example 2: 5⁻² = 1/5² = 1/25

Example 3: x⁻⁴ = 1/x⁴

Example 4: (3/4)⁻² = (4/3)² = 16/9

Example 5: 1/2⁻³ = 2³ = 8

8. Combining Multiple Laws

Often, we need to use more than one law to simplify an expression.

Example 1: Simplify (2² × 2³)⁴

First, inside parentheses: 2² × 2³ = 2^(2+3) = 2⁵
Then, (2⁵)⁴ = 2^(5×4) = 2²⁰

Example 2: Simplify (3²)³ × 3⁴

First, (3²)³ = 3^(2×3) = 3⁶
Then, 3⁶ × 3⁴ = 3^(6+4) = 3¹⁰

Example 3: Simplify (x³ × x²)⁴ ÷ x⁶

Inside: x³ × x² = x⁵
(x⁵)⁴ = x²⁰
x²⁰ ÷ x⁶ = x^(20-6) = x¹⁴

Example 4: Simplify (2⁻²)³ × 2⁵

(2⁻²)³ = 2^(-2×3) = 2⁻⁶
2⁻⁶ × 2⁵ = 2^(-6+5) = 2⁻¹ = 1/2

Solved Examples

Example 1 – Product Rule: Simplify 4⁷ × 4³

Solution: 4^(7+3) = 4¹⁰

Answer: 4¹⁰

Example 2 – Quotient Rule: Simplify 9⁸ ÷ 9⁵

Solution: 9^(8-5) = 9³

Answer: 9³

Example 3 – Power of a Power: Simplify (5³)⁴

Solution: 5^(3×4) = 5¹²

Answer: 5¹²

Example 4 – Power of a Product: Simplify (2x)⁵

Solution: 2⁵ × x⁵ = 32x⁵

Answer: 32x⁵

Example 5 – Power of a Quotient: Simplify (3/5)³

Solution: 3³ / 5³ = 27/125

Answer: 27/125

Example 6 – Combining Laws: Simplify (2³ × 2²)³

Solution: Inside: 2³ × 2² = 2⁵
(2⁵)³ = 2¹⁵

Answer: 2¹⁵

Example 7 – Negative Exponent: Simplify 3⁻⁴ × 3⁶

Solution: 3^(-4+6) = 3² = 9

Answer: 9

Common Mistakes to Avoid

Mistake 1 – Adding exponents when bases are different
2³ × 3² cannot be simplified to 6⁵ (wrong).
Correct understanding: Laws of exponents apply only to the same base.

Mistake 2 – Confusing product rule with power of a power
(2³)⁴ = 2¹², not 2⁷.
Correct understanding: Power of a power multiplies exponents; product rule adds exponents.

Mistake 3 – Forgetting to raise the coefficient
(3x)² = 9x², not 3x².
Correct understanding: Raise EVERY factor inside parentheses to the power.

Mistake 4 – Misapplying negative exponent rule
2⁻³ = 1/8, not -8. The negative exponent means reciprocal, not negative number.
Correct understanding: a^(-n) = 1/a^n (positive denominator).

Mistake 5 – Thinking a⁰ = 0
5⁰ = 1, not 0.
Correct understanding: Any non-zero number to the power zero equals 1.

Mistake 6 – Subtracting exponents in wrong order
a^m ÷ a^n = a^(m-n). Make sure denominator exponent is subtracted from numerator exponent.
Correct understanding: m – n, not n – m.

Quick Reference Summary – The Seven Laws

Remember:

• Same base for product/quotient rules
• Multiply exponents for power of a power
• Distribute exponent to all factors inside parentheses
• 0⁰ is undefined