If 4 = 10^2m and 9 = 10^2n, then 0.15 equals to :

1. Fundamental Laws of Indices and Exponents (basic)

Welcome to the first step of your quantitative aptitude journey! To master complex calculations, we must first understand Indices and Exponents—the mathematical shorthand for repeated multiplication. When we write 10³, it simply means 10 is multiplied by itself 3 times (10 × 10 × 10). Here, 10 is the base and 3 is the exponent or index. In competitive exams, you will often see large numbers like the 403 seats in the Uttar Pradesh Legislative Assembly Laxmikanth, M. Indian Polity, State Legislature, p.350; understanding indices allows us to break such numbers down into manageable prime factors or powers of 10. There are three fundamental laws you must memorize to manipulate these expressions quickly:

• The Product Law: When multiplying powers with the same base, we add the exponents. (aᵐ × aⁿ = aᵐ⁺ⁿ). For example, 10² × 10³ = 10⁵.
• The Quotient Law: When dividing powers with the same base, we subtract the exponents. (aᵐ / aⁿ = aᵐ⁻ⁿ). This is crucial for simplifying fractions.
• The Power Law: When raising a power to another power, we multiply the exponents. ((aᵐ)ⁿ = aᵐⁿ).

Beyond these, you must be comfortable with Negative and Zero Exponents. Any non-zero number raised to the power of 0 is always 1 (a⁰ = 1). Furthermore, a negative exponent like 10⁻¹ is simply a way to represent a reciprocal or a fraction, specifically 1/10¹ or 0.1. Just as the Indian Constitution allows for flexibility through simple parliamentary laws Indian Constitution at Work, NCERT 2025 ed., CONSTITUTION AS A LIVING DOCUMENT, p.201, these rules provide the flexibility to rewrite complex decimal values as clean, exponential expressions, making division and multiplication significantly faster during the exam.

Sources: Laxmikanth, M. Indian Polity, State Legislature, p.350; Indian Constitution at Work, NCERT 2025 ed., CONSTITUTION AS A LIVING DOCUMENT, p.201

2. Powers of 10 and Negative Exponents (basic)

At its heart, our entire mathematical world revolves around the number 10. This is known as the decimal system. Historically, this system has been the bedrock of organization—from the Harappan civilization, where higher denominations of weights followed a decimal pattern THEMES IN INDIAN HISTORY PART I, Bricks, Beads and Bones, p.16, to the military structures of Genghis Khan, who organized his army into divisions of 10s, 100s, and 1,000s Themes in world history, Nomadic Empires, p.69. In powers of 10, the exponent tells us how many times to multiply 10 by itself. For example, 10² is 10 × 10 = 100, and 10³ is 10 × 10 × 10 = 1,000. Essentially, the positive exponent tells you the number of zeros following the '1'.

While positive exponents help us represent vast numbers, negative exponents allow us to express very small values or decimals with precision. A common mistake is thinking a negative exponent makes the number itself negative; it does not! Instead, 10⁻¹ represents a reciprocal, meaning 1/10¹ or 0.1. Similarly, 10⁻² is 1/10² (which is 1/100 or 0.01). Every time the exponent decreases by one, we are effectively dividing the number by 10, moving the decimal point one place to the left.

Understanding how to manipulate these powers is a vital shortcut in quantitative aptitude. When you multiply powers of 10, you add the exponents (10² × 10³ = 10⁵). When you divide, you subtract the exponent of the divisor from the exponent of the dividend (10⁵ ÷ 10² = 10⁵⁻² = 10³). This rule remains consistent even with negative numbers: for instance, 1 / 10² can be rewritten as 10⁻². Mastery of these shifts allows you to convert complex decimals into manageable whole numbers and exponents, making division and multiplication much faster during an exam.

Power of 10	Fractional Value	Decimal Value
10¹	10/1	10
10⁰	1/1	1
10⁻¹	1/10	0.1
10⁻²	1/100	0.01

Sources: THEMES IN INDIAN HISTORY PART I, Bricks, Beads and Bones, p.16; Themes in world history, Nomadic Empires, p.69

3. Square Roots of Exponential Terms (intermediate)

Concept: Square Roots of Exponential Terms

4. Rational Numbers and Fractional Simplification (basic)

At its core, a rational number is any number that can be expressed in the form p/q, where both p and q are integers and q is not zero. In the context of the UPSC Civil Services Examination, you will often encounter these numbers disguised as decimals or percentages. Understanding how to transition between these forms is the secret to solving complex Quantitative Aptitude problems. For instance, in data interpretation or geography, a total is often represented as 100.00% Geography of India, Transport, Communications and Trade, p.48, which serves as the standard denominator for most percentage-to-fraction conversions.

To convert a decimal like 0.15 into a fraction, we look at the place value. Since the decimal extends to the hundredths place, 0.15 is mathematically equivalent to 15/100. In the study of energy resources, we find that the motor fuel fraction of distilled oil might be exactly 15 per cent Certificate Physical and Human Geography, Fuel and Power, p.271. However, to work efficiently with such numbers, we must perform fractional simplification. This involves finding the Highest Common Factor (HCF) of the numerator and the denominator and dividing both by it. For 15/100, the HCF is 5. Dividing both gives us the simplified fraction 3/20.

Moving a step further, an exceptional aspirant learns to express these simplified fractions in terms of prime factors or powers of 10. For example, the fraction 3/20 can be broken down into 3 / (2 × 10¹). This structural breakdown is incredibly useful when a question asks you to relate a decimal value to exponential variables. By mastering the transition from 0.15 → 15/100 → 3/20 → 3/(2 × 10), you gain the flexibility to solve equations involving logarithms or indices where the base is 10.

Sources: Geography of India, Transport, Communications and Trade, p.48; Certificate Physical and Human Geography, Fuel and Power, p.271

5. Scientific Notation and Orders of Magnitude (intermediate)

At its core, Scientific Notation is a standardized way to represent numbers that are either incredibly large or infinitesimally small, making them easier to compare and calculate. It expresses a number as a product of a coefficient (between 1 and 10) and a power of 10. For instance, instead of writing 0.000015, we might express it as 1.5 × 10⁻⁵. This concept is fundamentally linked to the decimal system, which has historical roots in India; for example, while the Harappans used binary systems for small weights, their higher denominations followed a strict decimal system to manage larger quantities THEMES IN INDIAN HISTORY PART I, Bricks, Beads and Bones, p.16.

An Order of Magnitude refers to the approximate measure of the size of a number, specifically the power of 10 that it represents. If one value is 100 times larger than another, we say it is two orders of magnitude greater (since 100 = 10²). This "power-based" thinking is common in global standards; just as we add or subtract hours from Greenwich Mean Time (GMT) to find local time zones Exploring Society: India and Beyond, Locating Places on the Earth, p.21, we use exponents to shift the decimal point to its correct "standard" position.

When working with these powers, we follow the Laws of Exponents. These rules are vital for simplifying complex expressions:

• Multiplication: 10ᵃ × 10ᵇ = 10ᵃ⁺ᵇ
• Division: 10ᵃ / 10ᵇ = 10ᵃ⁻ᵇ
• Negative Exponents: 10⁻ⁿ = 1/10ⁿ (representing values less than 1)

In many quantitative aptitude problems, you will need to convert decimals into these powers to simplify division. For example, 0.15 is 15/100, which can be broken down into 15 × 10⁻². By breaking numbers into their prime factors and powers of 10, we can solve complex algebraic identities without manual long division.

Sources: THEMES IN INDIAN HISTORY PART I, Bricks, Beads and Bones, p.16; Exploring Society: India and Beyond, Locating Places on the Earth, p.21

6. Substitution in Exponential Equations (exam-level)

In quantitative aptitude, Substitution in Exponential Equations is the art of 're-skinning' a numerical problem by expressing its components in terms of a common base. This technique is remarkably similar to the logic used in economics to solve simultaneous equations, where we hold certain variables constant (ceteris paribus) and solve for one variable in terms of another to find a complete solution Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.53. To master this, we first simplify our 'inventory' of given equations (e.g., if 9 = 10²ⁿ, then 3 = 10ⁿ) and then decompose our target decimal or fraction into its prime components (like 2, 3, or 5) that match our inventory. Once the target is broken down into a fraction, we apply the Laws of Exponents to finalize the substitution. The most critical rules for the UPSC exam are:

• Multiplication Rule: 10ᵃ × 10ᵇ = 10ᵃ⁺ᵇ
• Division Rule: 10ᵃ / 10ᵇ = 10ᵃ⁻ᵇ
• Power of a Power: (10ᵃ)ᵇ = 10ᵃᵇ

By replacing the integers in our fraction with their exponential equivalents, a complex division problem transforms into a simple subtraction of exponents. For instance, if you are looking for a decimal like 0.15, you first view it as 15/100, then simplify it to 3/(2 × 10), and finally substitute the exponential values of 3 and 2 to reach the result.

Sources: Macroeconomics (NCERT class XII 2025 ed.), Determination of Income and Employment, p.53

7. Solving the Original PYQ (exam-level)

This question is a brilliant application of the Laws of Exponents and Scientific Notation concepts you just mastered. In the CSAT, UPSC often hides simple integers inside powers to test your ability to simplify complex-looking expressions. Here, the first step is recognizing that 4 and 9 are perfect squares ($2^2$ and $3^2$). By taking the square root of both sides of the given equations, you bridge the gap between the base numbers (2 and 3) and the powers of 10, finding that $2 = 10^m$ and $3 = 10^n$. This is the "Eureka moment" where the building blocks of exponent manipulation come together to reveal the solution.

As your coach, I want you to look at $0.15$ not just as a decimal, but as a fraction: $15/100$, which simplifies to $3/20$. By further breaking $20$ down into $2 imes 10$, you can substitute your earlier findings. Substituting $3 = 10^n$ and $2 = 10^m$ into the fraction $3 / (2 imes 10^1)$ gives you $10^n / (10^m imes 10^1)$. Applying the rule of subtracting exponents during division, you arrive at the correct answer (D) 10^(n-m-1). Notice how systematic decomposition of the decimal makes the algebra straightforward. Following the principles in Just the Maths, we see that moving the decimal point is fundamentally an exercise in adjusting the exponent of base 10.

Watch out for common UPSC traps! Option (C) $10^{n-m+1}$ is a classic "sign error" trap where a student might mistakenly multiply by 10 instead of dividing. Option (A) $10^{2m-2n}$ is designed to catch those who forget to take the square root at the beginning, attempting to work directly with 4 and 9. The CSAT examiners rely on your haste to make these computational slips. Always double-check your sign when moving an exponent from the denominator to the numerator; that final "-1" in the exponent is crucial because the 10 was in the denominator.