Index

Introduction  

The index is important in mathematics, especially with exponential expressions. Often used interchangeably with "exponent" or "power," an index shows how many times a number is multiplied by itself. Whether it's algebra, scientific notation, or practical math modelling, understanding the index concept is essential.

In this guide, we will define an index, explain how it works in mathematical expressions, explore the laws of indices, and apply these rules through examples. Let’s dive into the world of indices in math.

Table of Contents  

• Index Definition
• Laws of Indices
• Rules of Indices
• Examples
• Real-Life Applications of Index
• Solved Examples
• Conclusion
• FAQs on Index

Index Definition

A number or a variable can have an index. The index tells us how many times to multiply a number its itself. In simple words, an index is a numerical value that indicates how many times a number (the base) is multiplied by itself.

It is written like this: a^m = a × a × a ×…× a (m times)

Here, a is called the base, and m is the index.

Key points:

• Base: The number being multiplied.

• Index (Exponent): Indicates the number of times the base is used as a factor.  

In simple words, the index shows how many times we should multiply the base by itself. It is a short way to write repeated multiplication and makes calculations easier.

• For example: 2³ = 2 × 2 × 2 = 8

• Here, 2 is the base and 3 is the index. 
  
  

Laws of Indices

Indices follow some simple rules that help us to do calculations easily. Here are the main rules explained in simple words

1. Any number to the power 0 to 1
No matter what the number is, if the index is 0, then the answer is always 1.

• • a⁰ = 1; example: 7⁰ = 1, 100⁰ = 1, 7⁰ = 1

2. Negative index means reciprocal.
A negative index means 1 divided by the number with a positive index.

• • a^-p = 1/a^p; example: 3^-2 =  1 / 3² = 1/9, 10⁻¹ = 1 / 10 = 0.1

3. Multiply the same base →  add powers
When multiplying numbers with the same base, add their powers.

• • a^p  × 1/a^ᵠ= a^p+ᵠ ;example: 2³ ×  2² = 2³⁺² = 2⁵ = 32

4. Divide same base →  subtract powers
When dividing numbers with the same base, subtract the power of the denominator from the numerator.

• • ^cc /a^ᵠ= a^p-ᵠ ;example: 5⁴ / 5² = 5⁴⁻² = 5² = 25
    

5. Power of a power → multiply powers.
When a number with an index is raised to another index, multiply the powers.

• • (a^p)^ᵠ= a^pᵠ ;example: (3²)⁴ = 3²·⁴ = 3⁸ = 6561
    

6. Multiply different bases with the same power → multiply bases.
When numbers have different bases but the same index, multiply the bases and raise them to that power.

• • a^p × b^p= ( a × b)^p ;example: 4³ × 2³ = (4 × 2)³ = 8³ = 512

7. Divide different bases, same power → Divide bases.
When numbers have different bases but the same index, divide the bases and raise to that power.

• • a^p / b^p= ( a / b)^p ;example: 9² / 3² = (9 / 3)² = 3² = 9

8. Fractional index → radical form.
If the index is a fraction, it can be written as a root.

• • a^{p / ᵠ}= ^ᵠ√a^p ;example: 16^1/2 = √16 = 4, 27^1/3 = ∛27 = 3

 

Rules of Indices

Let’s apply the laws of indices through examples:

• Product Rule: 3²× 3³ = 3²⁺³= 3⁵ = 243  

• Quotient Rule: 5⁴ ÷ 5² = 5^4-2 = 5² = 25  

• Power Rule: (2³)² = 2^3×2 = 2⁶ = 64  

• Zero Index Rule:  7⁰ = 1  

• Negative Index: 4^-2 = 1/4² = 1/16  

These rules simplify both algebraic and arithmetic operations.

 

Examples

Indices appear frequently in mathematical expressions. Here are some real-life examples:

• Scientific notation: 3.2 × 10⁵  

• Algebra:  x³ + 2x² - x + 5  

• Geometry: Area of a square: A = s²  

• Compound interest: A = P(1 + r)^t , These examples illustrate how useful and versatile indices are in maths.

Real-Life Applications of Index

• Engineering: Signal processing uses exponential scales.

• Finance: Compound interest formulas depend on indices.

• Science: Atomic decay and growth are modelled with exponents.

• Technology: Computer memory (bytes, kilobytes) is powers of 2.

• Astronomy: Distances are expressed in exponential form.

Understanding exponents and the laws of indices has great real-world relevance.

 

Solved Examples

Example 1:

Simplify 5² × 5³  

Ans: 5²⁺³ = 5⁵= 3125  

Example 2:

Evaluate (3²)³  

Ans: 3^2*3 = 3⁶ = 729  

Example 3:

Simplify 2⁵ / 2³  

Ans: 2^5-3 = 2² = 4.

Example 4:

Express 1/4 using a negative index.

Ans: 1/4 = 4^-1.  

Example 5:

Evaluate (2 × 3)² 

Ans: 2² × 3² = 4 × 9 = 36.

 

Conclusion

Mastering the concept of index is a key part of mathematics, especially with exponential expressions and algebraic simplifications. From understanding the definition to applying the laws and rules of indices, the use of indices in maths is broad and necessary.  

By grasping the meaning of exponents and applying real-world examples of indices, learners can discover new problem-solving strategies and strengthen their mathematical intuition. So the next time you see a small number raised above another, remember that’s the mighty index, making math both simple and powerful.

 

FAQs on Index

1. What is an index in mathematics?

Answer: In mathematics, an index (or exponent) is a number that indicates how many times the base number is multiplied by itself. For example, 2³ = 2×2×2 = 8.

2. What are the 4 types of index numbers?

Answer: The four main types of index numbers are:

• Price Index

• Quantity Index

• Value Index

• Volume Index

3. Why is the index 100?

Answer: Index numbers are often set to a base of 100 to simplify comparison. A value above or below 100 indicates a percentage change from the base period.

4. What is the concept of the index formula?

Answer: The index formula is used to measure the relative change of a value over time and is expressed as:
Index = (Current Value / Base Value) × 100

 

Understand index laws with stepwise examples from Orchids The International.