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WBB • Class 7 • Mathematics (গণিত প্রভা) • Ch 5

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Concept of Indices

Welcome to Chapter 5 "Concept of Indices" of the West Bengal Board (WBBSE) Class 7 Mathematics curriculum. Engineered with the TargetExams Gold-Standard 5-Step Pedagogy System, this module covers exponential notation, base and index definitions, the five fundamental laws of indices ($a^m \cdot a^n = a^{m+n}$, $a^m / a^n = a^{m-n}$, $(a^m)^n = a^{mn}$), the zero exponent theorem ($a^0 = 1$), negative indices ($a^{-n} = \frac{1}{a^n}$), prime base decomposition, and standard scientific notation for astronomical and atomic scales.

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Overview Core Concepts Formulas & Identities Solved Examples Common Traps Summary Self Check Take MCQ Test

🌍 The Legend of the Chessboard & The Power of Exponential Growth

How many grains of wheat can fit on a standard 64-square chessboard?

When the inventor of chess presented the game to the King of India, the monarch offered any reward. The inventor requested a seemingly modest boon: 1 grain of wheat on the 1st square, 2 on the 2nd, 4 on the 3rd, 8 on the 4th, doubling each consecutive square until all 64 squares were filled. The king readily agreed, assuming it was a modest request.

Royal treasurers quickly calculated that on the 64th square alone, the count was $2^{63}$ grains. The total grains required across the board was $2^{64} - 1 \approx 18{,}446{,}744{,}073{,}709{,}551{,}615$ grains—weighing roughly 460 billion metric tons, more than 1,000 times the modern world's annual grain production! This astonishing phenomenon is Exponential Growth. From nuclear chain reactions and viral propagation to supercomputing memory, indices govern the physics and mathematics of our universe.

Why This Chapter Matters

Before You Begin (Prerequisites)

Fluency in repeated multiplication and division of whole numbers.
Familiarity with Prime Factorization of composite numbers.
Understanding of fraction reciprocals and cancellation of common factors.

What You Will Learn (Core Objectives)

Identify base and index in exponential expressions and convert repeated products into exponential notation ($a^n$).
Apply the Product Law ($a^m \times a^n = a^{m+n}$) and Quotient Law ($a^m \div a^n = a^{m-n}$) across like bases.
Utilize the Power-of-a-Power Law ($(a^m)^n = a^{mn}$) and distributive laws for powers of products and quotients.
Rigourously demonstrate why $a^0 = 1$ for any non-zero base and evaluate negative exponents ($a^{-n} = \frac{1}{a^n}$).
Decompose composite numerical expressions into prime-base factors to simplify multi-tiered algebraic fractions.

Chapter Roadmap & Progression

1 1. Exponential Notation, Base & Ind...

2 2. Product and Quotient Laws for Co...

3 3. Power of a Power & Distributive...

4 4. The Zero Exponent Theorem & Nega...

5 5. Prime Base Decomposition & Stand...

Complete Concept Guide (100% Curriculum Coverage)

1. Exponential Notation, Base & Index Terminology

1. The Intuition

Writing out $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ is tedious and prone to counting errors. Exponential notation provides a concise, universal shorthand for repeated self-multiplication: 2 multiplied by itself 8 times is written compactly as $2^8$.

2. Formal Concept & Structure

For any real number $a$ multiplied by itself $n$ times:

$$\mathbf{a^n = \underbrace{a \times a \times a \times \dots \times a}_{n \text{ factors}}}$$

Base ($a$): The repeated factor being multiplied.
Index / Exponent / Power ($n$): The integer indicating how many times the base occurs as a factor.
Sign Behavior for Negative Bases:
• $(-a)^{\text{even power}} = +a^{\text{even}}$ (e.g., $(-2)^4 = (-2)(-2)(-2)(-2) = +16$)
• $(-a)^{\text{odd power}} = -a^{\text{odd}}$ (e.g., $(-2)^3 = (-2)(-2)(-2) = -8$)

3. Concrete Worked Example

Problem: Express $729$ in exponential form with base $3$, and $128$ with base $2$.

Step 1 (Prime factorize 729):

$729 = 3 \times 243 = 3 \times 3 \times 81 = 3 \times 3 \times 3 \times 27 = 3 \times 3 \times 3 \times 3 \times 9 = 3^6$

Step 2 (Prime factorize 128):

$128 = 2 \times 64 = 2 \times 2 \times 32 = 2^7$

Answer: $729 = \mathbf{3^6}$ (Base 3, Exponent 6); $128 = \mathbf{2^7}$ (Base 2, Exponent 7).

4. Pitfall & Examiner Trap

⚠️ Common Trap: Multiplying Base by Exponent
Students often evaluate $2^5$ as $2 \times 5 = 10$. This is fundamentally incorrect!
Correct: $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = \mathbf{32}$.

5. Why This Matters in Life

Computer memory architecture uses powers of 2 ($1\text{ KB} = 2^{10}\text{ Bytes} = 1024\text{ B}$, $1\text{ MB} = 2^{20}\text{ B}$, $1\text{ GB} = 2^{30}\text{ B}$).

2. Product and Quotient Laws for Common Bases

1. The Intuition

When multiplying $x^3$ by $x^4$, you are grouping 3 factors of $x$ with 4 factors of $x$: $(x \cdot x \cdot x) \cdot (x \cdot x \cdot x \cdot x) = x^7$. The total count of factors is $3 + 4 = 7$. Division performs the reverse: common factors cancel out from top and bottom, subtracting the powers!

2. Formal Concept & Laws

The Product Law: When multiplying powers with the same base, keep the base and add the exponents:
$$\mathbf{a^m \times a^n = a^{m+n}}$$

The Quotient Law: When dividing powers with the same base, keep the base and subtract the exponents:
$$\mathbf{a^m \div a^n = \frac{a^m}{a^n} = a^{m-n}} \quad (a \neq 0, m \ge n)$$

3. Concrete Worked Example

Problem: Simplify: $\frac{3^7 \times 3^6}{3^9}$

Step 1 (Product Law in numerator): $3^7 \times 3^6 = 3^{7 + 6} = 3^{13}$

Step 2 (Quotient Law): $\frac{3^{13}}{3^9} = 3^{13 - 9} = 3^4$

Step 3 (Evaluate): $3^4 = 3 \times 3 \times 3 \times 3 = \mathbf{81}$

Answer: $\mathbf{3^4}$ or $\mathbf{81}$

4. Pitfall & Examiner Trap

⚠️ Trap: Multiplying Exponents Instead of Adding
Writing $a^3 \times a^5 = a^{15}$ is a classic error.
Rule: In multiplication of common bases, powers add: $a^3 \times a^5 = a^{3 + 5} = \mathbf{a^8}$.

5. Why This Matters in Life

Decibel scale acoustical calculations and Richter earthquake magnitudes use logarithmic/exponential addition to measure energetic ratios.

3. Power of a Power & Distributive Laws over Products and Quotients

1. The Intuition

What does $(a^4)^3$ mean? It means three groups of $a^4$: $a^4 \times a^4 \times a^4 = a^{4 + 4 + 4} = a^{4 \times 3} = a^{12}$. When a power is raised to another power, the exponents multiply directly!

2. Formal Concept & Laws

Power of a Power Law: $\mathbf{(a^m)^n = a^{m \times n} = a^{mn}}$
Power of a Product Law: The power distributes across each factor in a product:
$$\mathbf{(a \times b)^n = a^n \times b^n = a^n b^n}$$
Power of a Quotient Law: The power distributes across both numerator and denominator:
$$\mathbf{\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}} \quad (b \neq 0)$$

Crucial Distinction: $(a^m)^n \neq a^{(m^n)}$
$(3^2)^3 = 3^{2 \times 3} = 3^6 = 729$, but $3^{(2^3)} = 3^8 = 6561$!

3. Concrete Worked Example

Problem: Simplify: $\frac{(2^4)^3 \times 9^2 \times 4}{8^3 \times 3^3}$

Step 1 (Convert all bases to primes 2 and 3):

• $(2^4)^3 = 2^{12}$

• $9^2 = (3^2)^2 = 3^4$

• $4 = 2^2$

• $8^3 = (2^3)^3 = 2^9$

Step 2 (Substitute and group):

$= \frac{2^{12} \times 3^4 \times 2^2}{2^9 \times 3^3} = \frac{2^{14} \times 3^4}{2^9 \times 3^3}$

Step 3 (Apply Quotient Law):

$= 2^{14 - 9} \times 3^{4 - 3} = 2^5 \times 3^1 = 32 \times 3 = \mathbf{96}$

Answer: $\mathbf{96}$

4. Pitfall & Examiner Trap

⚠️ Illegal Distribution: Distributing Powers Over Addition
$(a + b)^2 \neq a^2 + b^2$. Powers do NOT distribute across sums or differences.
Distribution holds strictly for multiplication and division: $(ab)^n = a^n b^n$ and $(a/b)^n = a^n/b^n$.

5. Why This Matters in Life

Scaling laws in biomechanics (e.g. why an elephant cannot jump: surface area scales as $L^2$ while volume and mass scale as $L^3$) depend on power laws.

4. The Zero Exponent Theorem & Negative Indices

1. The Intuition

Any non-zero quantity divided by itself equals 1: $\frac{a^5}{a^5} = 1$. By the Quotient Law of Indices: $\frac{a^5}{a^5} = a^{5 - 5} = a^0$. Equating the two expressions reveals the mathematical truth: $a^0 = 1$! To keep arithmetic consistent, any non-zero number raised to the power 0 must equal 1.

2. Formal Concept & Structure

The Zero Exponent Theorem:
For any non-zero real number $a$:
$$\mathbf{a^0 = 1} \quad (a \neq 0)$$ Examples: $4^0 = 1, \quad (-9)^0 = 1, \quad \left(\frac{5}{8}\right)^0 = 1, \quad 1{,}000{,}000^0 = 1$

The Negative Index Law:
A negative exponent represents the reciprocal raised to the positive exponent:
$$\mathbf{a^{-n} = \frac{1}{a^n}} \quad \text{and} \quad \mathbf{\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n} \quad (a, b \neq 0)$$

Undefined Case ($0^0$): Zero to the power zero ($0^0$) is mathematically indeterminate / undefined because $0/0$ cannot be assigned a unique value.

3. Concrete Worked Example

Problem: Evaluate: $\frac{2^0 + 3^0 + 4^0}{5^0} \times \left(\frac{3}{2}\right)^{-3}$

Step 1 (Evaluate zero exponents):

$2^0 = 1, 3^0 = 1, 4^0 = 1, 5^0 = 1 \implies \frac{1 + 1 + 1}{1} = 3$

Step 2 (Evaluate negative exponent):

$\left(\frac{3}{2}\right)^{-3} = \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}$

Step 3 (Multiply):

$= 3 \times \frac{8}{27} = \frac{8}{9}$

Answer: $\mathbf{\frac{8}{9}}$

4. Pitfall & Examiner Trap

⚠️ Examiner Trap: Believing $a^0 = 0$
Raising a base to power zero does NOT produce zero: $7^0 = 1$. The exponent zero indicates that equal factors cancelled out completely, leaving the multiplicative identity $1$.

5. Why This Matters in Life

Atomic physics (measuring Bohr radius $\approx 5.29 \times 10^{-11}\text{ m}$) and electronics (capacitance in microfarads $10^{-6}\text{ F}$) rely exclusively on negative powers of 10.

5. Prime Base Decomposition & Standard Scientific Notation

1. The Intuition

Astronomical and microscopic quantities contain excessive strings of zeroes. The speed of light is $300{,}000{,}000\text{ m/s}$. Writing it as $3 \times 10^8\text{ m/s}$ avoids errors and makes comparison immediate. This is Standard Scientific Notation.

2. Formal Concept & Structure

Powers of 10 Place-Value Expansion:

$$64{,}582 = (6 \times 10^4) + (4 \times 10^3) + (5 \times 10^2) + (8 \times 10^1) + (2 \times 10^0)$$

Scientific Notation Form:

Expressing numbers as $\mathbf{m \times 10^k}$, where $1 \le m < 10$ and $k \in \mathbb{Z}$.

Distance from Earth to Sun: $149{,}600{,}000\text{ km} = \mathbf{1.496 \times 10^8\text{ km}}$
Mass of the Earth: $\mathbf{5.972 \times 10^{24}\text{ kg}}$

3. Concrete Worked Example

Problem: Express $1440$ as a product of powers of prime factors.

Step 1 (Decompose into prime factors):

$1440 = 2 \times 720 = 2^2 \times 360 = 2^3 \times 180 = 2^4 \times 90 = 2^5 \times 45$

$45 = 3 \times 15 = 3^2 \times 5^1$

Step 2 (Combine):

$= \mathbf{2^5 \times 3^2 \times 5^1}$

Answer: $1440 = \mathbf{2^5 \times 3^2 \times 5^1}$

4. Pitfall & Examiner Trap

⚠️ Common Mistake: Cancelling unlike composite bases
Attempting to cancel between $6^4$ and $2^2$ without factoring $6 = 2 \times 3$.
Always factorize composite bases first: $6^4 = (2 \times 3)^4 = 2^4 \times 3^4$, then cancel common prime factors cleanly.

5. Why This Matters in Life

Astrophysicists and space agencies (ISRO, NASA) calculate orbital trajectories and planetary masses using scientific powers of 10 exclusively.

Key Formulas, Identities & Theorems

Product Law of Indices

$$a^m \times a^n = a^{m+n}$$

Add powers when bases are identical

Quotient Law of Indices

$$a^m \div a^n = a^{m-n}$$

Subtract powers for identical bases (a \neq 0)

Power of a Power Law

$$(a^m)^n = a^{m \times n} = a^{mn}$$

Multiply exponents directly

Power of a Product

$$(ab)^n = a^n \times b^n$$

Exponent distributes over factors

Power of a Quotient

$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

Exponent distributes over division (b \neq 0)

Zero Exponent Law

$$a^0 = 1 \quad (a \neq 0)$$

Any non-zero base to power 0 equals 1

Negative Exponent Law

$$a^{-n} = \frac{1}{a^n}, \quad \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$$

Negative power inverts the base

Conceptual Solved Examples & Case Studies

Example 1

Simplify and express in exponential form: $\frac{2^3 \times 3^4 \times 4}{3 \times 32}$

Step-by-Step Solution:

Express $4$ and $32$ with base $2$: $4 = 2^2$ and $32 = 2^5$.
$= \frac{2^3 \times 3^4 \times 2^2}{3^1 \times 2^5} = \frac{2^{3+2} \times 3^4}{2^5 \times 3^1} = \frac{2^5 \times 3^4}{2^5 \times 3^1}$
$= 2^{5-5} \times 3^{4-1} = 2^0 \times 3^3 = 1 \times 27 = \mathbf{27}$
Answer: $\mathbf{27}$ or $\mathbf{3^3}$

Example 2

Simplify: $\frac{(2^5)^2 \times 7^3}{8^3 \times 7}$

Step-by-Step Solution:

Step 1: $(2^5)^2 = 2^{10}$.
Step 2: $8^3 = (2^3)^3 = 2^9$.
Expression: $\frac{2^{10} \times 7^3}{2^9 \times 7^1} = 2^{10-9} \times 7^{3-1} = 2^1 \times 7^2 = 2 \times 49 = \mathbf{98}$
Answer: $\mathbf{98}$

Example 3

Solve for $x$: $3^{2x - 1} = 243$

Step-by-Step Solution:

Express $243$ as a power of $3$: $243 = 3^5$.
Since $3^{2x - 1} = 3^5$, equating exponents gives:
$2x - 1 = 5 \implies 2x = 6 \implies x = \mathbf{3}$
Answer: $x = \mathbf{3}$

Common Misconceptions & Examiner Traps

Common Misconception

Adding bases or multiplying powers when multiplying common bases ($2^3 \times 2^4 = 2^{12}$ or $4^7$).

Scientific Reality & Correction

The base remains unchanged, while powers add: $2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$.

Common Misconception

Confusing $(a^m)^n$ with $a^{(m^n)}$.

Scientific Reality & Correction

$(2^3)^2 = 2^6 = 64$, whereas $2^{(3^2)} = 2^9 = 512$.

Common Misconception

Writing $a^0 = 0$.

Scientific Reality & Correction

By definition of division of identical quantities, $a^0 = 1$ for any non-zero real number $a$.

Visual Learning & Conceptual Map

WBBSE Mathematics Foundations: The Fundamental Laws of Indices & Power Navigator

Anatomy of exponential expressions and the four cardinal transformation laws

Exponential Anatomy

aⁿ

↑ Base (Factor) ↑ Exponent (Count)

Core Definition: $a^n$ represents the base $a$ multiplied by itself $n$ times.
• $a = \text{Base (repeated entity)}$
• $n = \text{Exponent/Power (multiplication tally)}$
• $a^0 = 1 \quad (a \neq 0)$ and $a^{-n} = \frac{1}{a^n}$

1. Product Law

$a^m \times a^n = \mathbf{a^{m+n}}$

Add exponents

2. Quotient Law

$a^m \div a^n = \mathbf{a^{m-n}}$

Subtract exponents

3. Power of a Power

$(a^m)^n = \mathbf{a^{mn}}$

Multiply exponents

4. Zero & Negative

$a^0 = \mathbf{1}, \quad a^{-n} = \mathbf{\frac{1}{a^n}}$

Non-zero bases

Chapter Summary & 10 Key Takeaways

Takeaway 1

Exponential Notation ($a^n$): Shorthand for $a$ multiplied by itself $n$ times ($a$ is base, $n$ is index).

Takeaway 2

Product Law: $a^m \times a^n = a^{m+n}$ (add powers for common bases).

Takeaway 3

Quotient Law: $a^m \div a^n = a^{m-n}$ ($a \neq 0$, subtract powers for common bases).

Takeaway 4

Power of a Power: $(a^m)^n = a^{mn}$ (multiply exponents).

Takeaway 5

Distributive Laws: $(ab)^n = a^n b^n$ and $(a/b)^n = a^n / b^n$.

Takeaway 6

Zero Exponent: $a^0 = 1$ for any non-zero real base $a$.

Takeaway 7

Negative Exponent: $a^{-n} = 1/a^n$ and $(a/b)^{-n} = (b/a)^n$.

Takeaway 8

Scientific Notation: Expressing quantities as $m \times 10^k$ where $1 \le m < 10$.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

Evaluate: $(3^0 + 2^0) \times 5^2$

Reveal Answer & Explanation

Answer: $(1 + 1) \times 25 = 2 \times 25 = \mathbf{50}$
Any non-zero quantity raised to power zero equals 1.

Simplify: $\frac{2^8 \times a^5}{4^3 \times a^3}$

Reveal Answer & Explanation

Answer: Convert $4^3 = (2^2)^3 = 2^6$. Then $2^{8-6} \times a^{5-3} = 2^2 \times a^2 = \mathbf{4a^2}$.
Convert composite base 4 to prime base 2.

Find the value of $x$ if $2^{x+3} = 64$.

Reveal Answer & Explanation

Answer: $64 = 2^6 \implies x + 3 = 6 \implies x = \mathbf{3}$.
Express 64 as a power of 2.

Evaluate: $\left(\frac{2}{3}\right)^{-2} \times \left(\frac{4}{9}\right)^2$

Reveal Answer & Explanation

Answer: $\left(\frac{3}{2}\right)^2 \times \frac{16}{81} = \frac{9}{4} \times \frac{16}{81} = \mathbf{\frac{4}{9}}$
Invert the base to make the negative power positive: $(2/3)^{-2} = (3/2)^2 = 9/4$.

Express $1080$ in prime-base exponential form.

Reveal Answer & Explanation

Answer: $1080 = \mathbf{2^3 \times 3^3 \times 5^1}$
Divide repeatedly by prime factors 2, 3, and 5.

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