Power, Indices, and Exponents: These three terms are closely related and often used interchangeably in mathematics.
The exponent is the number that indicates how many times the base is multiplied by itself. In $a^n$, $a$ is called the base and $n$ is called the index or exponent .
For any non-zero real number $a$ and natural number $n$:
$$a^n = a × a × a × \dots × a \quad (n times)$$
$$a^1 = a, \quad a^2 = a × a, \quad a^3 = a × a × a$$
$$Example: 2^4 = 2 × 2 × 2 × 2 = 16$$
Special Powers & Boundary Conditions:
Zero Power: $a^0 = 1$ for any $a \ne 0$. (E.g. $4^0 = 1$). Note: $0^0$ is not defined and is called an indeterminate form. Negative Power: For positive $n$, $a-n = 1 a^n -1 = 1 2 -2 = 1 4 -3 = 1 8 Fractional Power: $a1/n = \sqrt[n]{a}$. E.g. $91/2 = √9 = 3$, $81/3 = \sqrt[3]{8} = 2$. More generally, $am/n = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$.
Let $a$ and $b$ be non-zero real numbers, and let $m$ and $n$ be rational numbers. The operations of exponents satisfy the following governing laws:
1. Product Law:
$$a^m · a^n = am+n $$
Example: $2^3 · 2^2 = 23+2 = 2^5 = 32$
2. Quotient Law:
$$a^m a^n m-n $$
Example: $2^3 2^2 3-2 = 2$, $2^2 2^3 2-3 = 2-1 = 1 2
3. Power of a Power Law:
$$(a^m)^n = amn $$
Example: $(2^2)^3 = 22 × 3 = 2^6 = 64$
4. Power of a Product Law:
$$(ab)^m = a^m b^m$$
Example: $3^2 · 4^2 = (3 · 4)^2 = 12^2 = 144$
5. Power of a Quotient Law:
$$(a b a^m b^m Example: $6^2 2^2 6 2
Example 4 (a): Simplify $4(6
2 3 )
3 2 + (4^3)
4 3 + 2 × (3^5)
1 5 $.
Show Step-by-Step Solution
Apply power of a power law $(a^m)^n = amn $ to each term:2 3 3 2 3 4 1 5
Example 4 (b): Simplify $
√1 4 + (0.01)
-1/2 - 27
2/3 $.
Show Step-by-Step Solution
Evaluate each individual term:√1 4 = 1 2 -1/2 = (10-2 )-1/2 = 10(-2) × (-1/2) = 10^1 = 10$2/3 = (3^3)2/3 = 33 × (2/3) = 3^2 = 9$1 2 1 2 3 2
Example 4 (c): Simplify $
5n+2 - 6 · 5n+1 13 · 5^n - 2 · 5n+1 $.
Show Step-by-Step Solution
Factor out common base power $5^n$ from both numerator and denominator:5n+2 - 6 · 5n+1 13 · 5^n - 2 · 5n+1 5^n(5^2 - 6 × 5^1) 5^n(13 - 2 × 5^1) 25 - 30 13 - 10 -5 3 5 3
Example 4 (d): Simplify $16 - 3 × 2^2 + ((
6 9 )^2)
1/2 $.
Show Step-by-Step Solution
• Evaluate $3 × 2^2 = 3 × 4 = 12$6 9 1/2 = 6 9 2 3 2 3 2 3 14 3
Next: Check Your Progress
Simplify the following expressions using the laws of exponents and indices. Click each solution toggle to verify your working.
Simplify the expression: $a = x1 2 1 3
Show Step-by-Step Solution
Step 1: Apply the Product Law: $a^m · a^n = am+n $.1 2 1 3 1 2 1 3 Step 2: Find the common denominator for the exponents: $1 2 1 3 3 + 2 6 5 6 Answer: $\mathbf{a = x5/6 }$.
Simplify the expression: $a = x^2 · x2 3
Show Step-by-Step Solution
Step 1: Apply Product Law: $a^m · a^n = am+n $.2 3 2 + 2 3 = x6+2 3 8 3 Answer: $\mathbf{a = x8/3 }$.
Simplify: $a = (x2 3
Show Step-by-Step Solution
Step 1: Apply Power of a Power Law: $(a^m)^n = amn $.2 3 2 3 2 × 2 = x^4$$
Answer: $\mathbf{a = x^4}$.
Simplify: $a = x^3 y^4 x y^2
Show Step-by-Step Solution
Step 1: Apply Quotient Law $a^m a^n m-n $ to each variable base independently:x^3 x^1 3-1 = x^2$$
$$y^4 y^2 4-2 = y^2$$
Answer: $\mathbf{a = x^2 y^2 = (xy)^2}$.
Next: Worksheet
Printable classroom and homework worksheet for Indices & Exponents . Selected questions directly from the CBSE Applied Mathematics support material.
Section A: Conceptual & Objective Drill
3 Questions
Evaluate the product: $65610.14 × 65610.11 $.
Show Model Solution
By Product Law: $65610.14 + 0.11 = 65610.25 = 65611/4 $.1/4 = 38/4 = 3^2 = \mathbf{9}$.
Find the exact value of the difference: $7-14 - 7-15 $.
Show Model Solution
Factor out the smaller power $7-15 $:-14 - 7-15 = 7-15 (7^1 - 1) = 7-15 (6) = \mathbf{6 × 7-15 }$.
Evaluate: $(6561)1/2 + (6561)1/4 + (6561)1/8 $.
Show Model Solution
• $(6561)1/2 = √6561 = 81$1/4 = √81 = 9$1/8 = √9 = 3$
Section B: Short Numerical Problems
2 Questions • 2 Marks Each
If $3n+4 - 3n+2 = 8$, find the value of $n$.
Show Model Solution
Factor out $3n+2 $:n+2 (3^2 - 1) = 8 \implies 3n+2 (8) = 8 \implies 3n+2 = 1$$
Since $3^0 = 1$: $n + 2 = 0 \implies \mathbf{n = -2}$.
If $5a+b = 5 × 25 × 125$, what is the value of $(a+b)^2$?
Show Model Solution
Write RHS in powers of 5: $5^1 × 5^2 × 5^3 = 51+2+3 = 5^6$.
Section C: High-Yield Applied & Advanced Problems
2 Questions
If $36120 = (36 × x)40 $, determine the value of $x$.
Show Model Solution
Rewrite LHS using power of a power: $36120 = (36^3)40 $.40 = (36 × x)40 \implies 36^3 = 36 × x$.36^3 36
Evaluate the exact radical expression: $[3-2 - 5-2 3-2 + 5-2 1/2 $.
Show Model Solution
Convert negative exponents: $3-2 = 1 9 -2 = 1 25 1 9 1 25 25 - 9 225 16 225 1 9 1 25 25 + 9 225 34 225 16/225 34/225 16 34 8 17 √8 17 = √8 √17 2√2 √17
Next: Revision Sheet
Essential formulas, laws, and examination tips for rapid last-minute revision.
Law / Rule
Formula
Key Condition / Notes
Product Law $$a^m · a^n = am+n $$ Same base, exponents add Quotient Law $$a^m a^n m-n $$ $a \ne 0$; subtract exponents Power of Power $$(a^m)^n = amn $$ Multiply exponents Product to Power $$(ab)^m = a^m b^m$$ Distributes across factors Quotient to Power $$(a b a^m b^m $b \ne 0$ Zero Exponent $$a^0 = 1$$ $a \ne 0$; $0^0$ is indeterminate Negative Exponent $$a-n = 1 a^n $a \ne 0$ Rational / Fractional $$am/n = (\sqrt[n]{a})^m$$ $n$th root raised to power $m$
$(a+b)^n \ne a^n + b^n$. Exponents DO NOT distribute over addition or subtraction!
$a-n \ne -a^n$. A negative exponent indicates a reciprocal, NOT a negative value (e.g. $2-3 = 1/8$, not $-8$).
Always reduce composite bases (like 6, 8, 27, 36) to prime factor powers ($2^p, 3^q$) before simplifying algebraic expressions!
Next: 10-Q Board Mock Test
Exercise 1.2 Diagnostic Board-Level Mock Examination
Exact pattern test covering Objective MCQs, Assertion-Reason, Short Answer Types, Long Answer, and a Case-Based Study.
Section A: Objective & Assertion-Reason (Q1 – Q4)
4 Questions • 1 Mark Each
What is the value of $65610.14 × 65610.11 $?
Solution: By product law: $65610.14 + 0.11 = 65610.25 = 65611/4 = (3^8)1/4 = 3^2 = 9$.
If $(√3 )^n = 6561$, then what is the value of $(n)1/2 $?
(A) 64
(B) $64√3 $
(C) $16√3 $
(D) 4
Solution: $(√3 )^n = 3n/2 $. Also $6561 = 3^8$. So $n/2 = 8 \implies n = 16$. Then $(n)1/2 = √16 = 4$.
What is the value of $(7-14 - 7-15 )$?
(A) $6 × 7-15 $
(B) $6 × 7-14 $
(C) $7 × 7-15 $
(D) $7 × 7-14 $
Solution: $7-14 - 7-15 = 7-15 (7^1 - 1) = 6 × 7-15 $.
Assertion (A): For any non-zero real number $a$, $a^0 = 1$.Reason (R): By the quotient law of exponents, $a^m a^m m-m = a^0 = 1$.
(A) Both A and R are true, and R is the correct explanation of A.
(B) Both A and R are true, but R is not the correct explanation of A.
(C) A is true, but R is false.
(D) A is false, but R is true.
Solution: Both A and R are true, and R provides the direct formal proof of why non-zero bases raised to power zero equal 1.
Section B: Short Answer Type I (Q5 – Q6)
2 Questions • 2 Marks Each
If $3n+4 - 3n+2 = 8$, find the value of $n$.
Show Marking Scheme & Solution (2 Marks)
Step 1 (1 Mark): Factor out $3n+2 $:n+2 (3^2 - 1) = 8 \implies 3n+2 (8) = 8 \implies 3n+2 = 1$.Step 2 (1 Mark): Since $3^0 = 1$, equate exponents: $n + 2 = 0 \implies \mathbf{n = -2}$.
Simplify the expression: $√1 4 + (0.01)-1/2 - 272/3 $.
Show Marking Scheme & Solution (2 Marks)
Step 1 (1 Mark): Simplify each term:√1/4 = 1/2$-1/2 = (10-2 )-1/2 = 10^1 = 10$2/3 = (3^3)2/3 = 3^2 = 9$.Step 2 (1 Mark): Combine: $1/2 + 10 - 9 = 1/2 + 1 = \mathbf{3/2}$.
Section C: Short Answer Type II (Q7 – Q8)
2 Questions • 3 Marks Each
Simplify $5n+2 - 6· 5n+1 13· 5^n - 2· 5n+1
Show Marking Scheme & Solution (3 Marks)
Step 1 (1 Mark): Factor $5^n$ from numerator and denominator:5^n(5^2 - 6 × 5) 5^n(13 - 2 × 5) Step 2 (1 Mark): Cancel $5^n$ since $5^n \ne 0$: $25 - 30 13 - 10 Step 3 (1 Mark): Simplify: $-5 3 5 3
If $36120 = (36 × x)40 $, determine the numerical value of $x$ using exponent laws.
Show Marking Scheme & Solution (3 Marks)
Step 1 (1 Mark): Rewrite LHS: $36120 = (36^3)40 $.Step 2 (1 Mark): Equate bases since powers are equal: $36^3 = 36 × x$.Step 3 (1 Mark): Solve for $x$: $x = 36^3 36
Section D: Long Answer Type (Q9)
1 Question • 4 Marks
Evaluate $[3-2 - 5-2 3-2 + 5-2 1/2 $ and simplify $2n+1 + 2^n 2n+2 - 2n+1
Show Marking Scheme & Solution (4 Marks)
Part 1 [2 Marks]: $3-2 = 1/9, 5-2 = 1/25$. Numerator $= 1/9 - 1/25 = 16/225$. Denominator $= 1/9 + 1/25 = 34/225$. Ratio $= 16/34 = 8/17$. Square root $= √8/17 = \mathbf{2√2 √17 Part 2 [2 Marks]: Factor $2^n$: $2^n(2 + 1) 2^n(4 - 2) 3 2
Section E: Case-Based / Source-Based Problem (Q10)
1 Case Study • 4 Marks
Case Background: In a biological research facility, a bacterial culture follows an exponential growth model given by $N(t) = 500 · 2t/3 $, where $t$ is measured in hours.
(i) [1 Mark]: Find the initial bacteria population at $t = 0$.
Show Solution for (i)
[1 Mark]: At $t = 0$: $N(0) = 500 · 2^0 = 500 × 1 = \mathbf{500}$ bacteria.
(ii) [1 Mark]: Determine the bacteria population after 6 hours.
Show Solution for (ii)
[1 Mark]: At $t = 6$: $N(6) = 500 · 26/3 = 500 · 2^2 = 500 × 4 = \mathbf{2000}$ bacteria.
(iii) [2 Marks]: Calculate the time $t$ in hours when the bacteria count will reach 16,000.
Show Solution for (iii)
[2 Marks]: Set $N(t) = 16000$: $500 · 2t/3 = 16000 \implies 2t/3 = 32$. Since $32 = 2^5$: $t/3 = 5 \implies \mathbf{t = 15 hours}$.