Exponent Rules Lesson: Definition, Rules, and Examples

Lesson Overview

• What Are Exponents and their Law?
• Laws and Rules of Exponents
• Exponent Rules Chart
• Exponent laws provide a framework for performing operations like multiplication and division with exponents, making complex calculations more manageable.
• Examples of Exponent Rules

Exponents are a fundamental concept in mathematics that simplify how we express and work with numbers, especially very large or very small numbers. The scope of exponents is crucial for a wide range of mathematical operations. 

The rules of exponents provide a clear framework for manipulating and simplifying expressions involving these powerful little numbers.

What Are Exponents and their Law?

Exponents provide a shorthand notation for repeated multiplication. Instead of writing out a number multiplied by itself many times, we use a raised number called an exponent to indicate how many times the multiplication occurs.  

For example, 5 x 5 x 5 x 5 can be written concisely as  5⁴.

• The base is the number being multiplied (5 in this example).
• The exponent or power is the number of times the base is multiplied by itself (4 in this example). 
  

Exponent laws are a set of rules that govern how expressions with exponents are simplified and combined.

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Laws of Exponents! Algebra Trivia Quiz

Laws and Rules of Exponents

Let us now learn about each exponent rule in detail and understand how they work-

1. Product of Powers

This law applies when multiplying exponential terms with the same base. To simplify, you add the exponents.

xᵐ X  xⁿ = x⁽ᵐ⁺ⁿ⁾

Multiplying xᵐ by xⁿ  is the same as multiplying 'x' by itself 'm' times, and then multiplying 'x' by itself 'n' more times. This results in 'x' being multiplied by itself a total of (m + n) times.

Example:

5³ X 5¹ = 5⁽³⁺¹⁾ = 5⁴ = 625

2. Quotient of Powers

This law applies when dividing exponential terms with the same base. To simplify, you subtract the exponents.

xᵐ / xⁿ = x⁽ᵐ⁻ⁿ⁾

Dividing xᵐ by xⁿ is essentially the same as canceling out 'n' of the 'x' terms in the numerator. This results in 'x' being multiplied by itself (m - n) times.

Example:

10⁵ / 10² = 10⁽⁵⁻²⁾ = 10³ = 1000

3. Power of a Power

This law applies when raising an exponential term to another power. To simplify, you multiply the exponents.

(xᵐ)ⁿ = x⁽ᵐⁿ⁾

Raising xᵐ to the power of n means multiplying xᵐ by itself 'n' times.  Because xᵐ represents 'x' multiplied by itself 'm' times, the result is 'x' being multiplied by itself a total of (m ⋅ n) times.

Example:

(2³)⁴ = 2⁽³⋅⁴⁾ = 2¹² = 4096

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Exponent Rules Quiz questions set 2

4. Power of a Product

This law applies when raising a product (multiplication of terms) to a power. The exponent is distributed to each factor in the product.

(xy)ⁿ = xⁿyⁿ

Raising (xy) to the power of 'n' means multiplying (xy) by itself 'n' times. Because multiplication is associative and commutative, this is equivalent to multiplying 'x' by itself 'n' times and 'y' by itself 'n' times.

Example:

(3x)² = 3² ⋅ x² = 9x²

5. Power of a Quotient

This law applies when raising a quotient (division of terms) to a power. The exponent is distributed to both the numerator and the denominator.

(x/y)ⁿ = xⁿ/yⁿ

Raising (x/y) to the power of 'n' means multiplying (x/y) by itself 'n' times. This is the same as multiplying 'x' by itself 'n' times in the numerator and 'y' by itself 'n' times in the denominator.

Example:

(2/5)³ = 2³/5³ = 8/125

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Power and Exponent Laws

6. Zero Exponent

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

x⁰ = 1 (where x ≠ 0)

The concept of a zero exponent can be understood using the quotient of powers rule. Dividing xᵐ by xᵐ (where 'm' is any number) results in x⁽ᵐ⁻ᵐ⁾ = x⁰.  Since any number divided by itself equals 1, this demonstrates that x⁰ = 1.

Example:

100⁰ = 1

7. Negative Exponent

A number raised to a negative exponent is equal to 1 divided by that number raised to the positive exponent.

x⁻ⁿ = 1/xⁿ

This can also be understood using the quotient of powers rule. Consider x²/x⁵. This is x⁽²⁻⁵⁾ = x⁻³. It can also be written as (x ⋅ x) / (x ⋅ x ⋅ x ⋅ x ⋅ x), which simplifies to 1/(x ⋅ x ⋅ x) or 1/x³.

Example:

2⁻³ = 1/2³ = 1/8

8. Fractional Exponent

A fractional exponent indicates a root. The numerator of the fraction represents the power, and the denominator represents the root.

x⁽ᵐ/ⁿ⁾ = ⁿ√(xᵐ) (nth root of xᵐ)

This law connects exponents with radicals (roots). It's a way to express roots using exponents.

Example:

9⁽¹/ ²⁾ = √9 = 3 (square root of 9)

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Laws of Exponents Review Quiz

Exponent Rules Chart

Exponent laws provide a framework for performing operations like multiplication and division with exponents, making complex calculations more manageable.

Examples of Exponent Rules

1. Simplify 7⁵ X 7²
   Solution:
   

• Apply the rule: xᵐ X xⁿ = x⁽ᵐ⁺ⁿ⁾
• Add the exponents: 7⁵ X 7² = 7⁽⁵⁺²⁾
• Simplify: 7⁷
  

2. Simplify a¹⁰ / a⁶

Solution:

• Apply the rule: xᵐ / xⁿ = x⁽ᵐ⁻ⁿ⁾
• Subtract the exponents: a¹⁰ / a⁶ = a⁽¹⁰⁻⁶⁾
• Simplify: a⁴
  

3.  Simplify (x³)⁴
   

Solution:

• Apply the rule: (xᵐ)ⁿ = x⁽ᵐⁿ⁾
• Multiply the exponents: (x³)⁴ = x⁽³⋅⁴⁾
• Simplify: x¹²
  

4. Simplify (2y)⁵
   Solution:
   

• Apply the rule: (xy)ⁿ = xⁿyⁿ
• Distribute the exponent: (2y)⁵ = 2⁵ X y⁵
• Simplify: 32y⁵
  

5. Simplify (x⁴ X x⁻²) / x³
   Solution:
   

• Apply the product of powers rule: x⁴ X x⁻² = x⁽⁴⁻²⁾ = x²
• Apply the quotient of powers rule: x² / x³ = x⁽²⁻³⁾
• Simplify: x⁻¹ = 1/x

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Test your understanding of exponents quiz!