Class 9 Mathematics Chapter 10: Exponents and Powers Previous Year Questions (2020–2025)

Reading Chapter 10 theory a third time rarely improves your score. Instead, solving past papers forces your brain to: 1. **Recognize Question Patterns**: CBSE repeats question structures. Once you solve 3–4 similar 3-mark questions, you can spot the approach instantly. 2. **Identify Your Blind Spots**: A PYQ reveals gaps faster than your textbook. For example, many students memorize a^m × a^n = a^(m+n) but freeze when asked to simplify (2³)^(-2) or express 0.00047 in scientific notation. 3. **Practice Under Time Pressure**: Board exams allow ≈90 seconds per mark. Solving past papers trains speed—you learn which steps to write versus skip. 4. **Build Answer-Writing Confidence**: Past paper solutions show how examiners expect working to be laid out. A correct answer with no steps loses marks; a complete but messier answer often scores full marks. 5. **Spot Recurring Topics**: Our analysis of 2020–2025 papers shows laws of exponents appear in 95% of tests, negative exponents in 80%, and expressing large/small numbers in 70%. Knowing these odds helps you allocate study time wisely. This section guide provides 13 questions spanning all three mark categories, each with solutions written in exam-standard format. Work through them sequentially, and you will retain mastery far longer than from passive reading.

One-mark questions test quick recall and fundamental skills. They often appear as fill-in-the-blank, MCQ, or very short answer. **Question 1**: Simplify: 2⁵ × 2³ *Answer*: 2⁸ = 256 *Explanation*: Apply the law a^m × a^n = a^(m+n). Here, 5 + 3 = 8. **Question 2**: Write 3⁻² in the form of a positive exponent. *Answer*: 1/3² = 1/9 *Explanation*: The law a^(-n) = 1/(a^n) converts negative exponents to fractions. Always state the final numerical answer: 1/9. **Question 3**: Simplify: (5²)³ *Answer*: 5⁶ = 15,625 *Explanation*: Power of a power rule: (a^m)^n = a^(m×n). Here, 2 × 3 = 6. Many students forget to multiply the exponents. **Question 4**: Express 8,00,000 in the form a × 10^n, where 1 ≤ a < 10. *Answer*: 8 × 10⁵ *Explanation*: Move the decimal point 5 places to the left. Standard form requires the coefficient to be between 1 and 10 (inclusive of 1). The exponent equals the number of places moved. **Question 5**: Simplify: (2/3)⁻¹ *Answer*: 3/2 *Explanation*: When a fraction has a negative exponent, invert the fraction and drop the negative sign: (a/b)^(-n) = (b/a)^n. So (2/3)⁻¹ = (3/2)¹ = 3/2. *Tip*: These 1-mark questions appear at the start of papers. Solving them correctly and quickly (under 30 seconds each) builds confidence and guarantees at least 5 marks.

Three-mark questions require multiple steps, showing your understanding of two or more concepts combined. **Question 1**: Simplify (2⁻¹ + 3⁻¹)⁻¹ *Solution*: Step 1: 2⁻¹ = 1/2, 3⁻¹ = 1/3 Step 2: 1/2 + 1/3 = 3/6 + 2/6 = 5/6 Step 3: (5/6)⁻¹ = 6/5 *Answer*: 6/5 or 1.2 *Concept*: Negative exponents, fraction addition, inverting fractions. **Question 2**: If 2^x = 32, find the value of x. Then simplify 2^(x-2) × 2^(3-x). *Solution*: Step 1: 2^x = 32 = 2⁵, so x = 5 Step 2: 2^(5-2) × 2^(3-5) = 2³ × 2⁻² = 2^(3-2) = 2¹ = 2 *Answer*: x = 5; final simplified form = 2 *Concept*: Solving exponential equations, laws of exponents for products. **Question 3**: Express 0.0000089 in scientific notation. *Solution*: Step 1: Move the decimal 6 places to the right to get 8.9 Step 2: The exponent = –6 (because we moved right, it's negative) Step 3: Answer = 8.9 × 10⁻⁶ *Verification*: 8.9 × 10⁻⁶ = 8.9/1,000,000 = 0.0000089 ✓ *Concept*: Scientific notation for small numbers, negative powers of 10. **Question 4**: Simplify: (a³b²) / (a⁻¹b³) where a ≠ 0, b ≠ 0 *Solution*: Step 1: (a³b²) / (a⁻¹b³) = a³ / a⁻¹ × b² / b³ Step 2: a^(3-(-1)) × b^(2-3) = a⁴ × b⁻¹ Step 3: a⁴/b *Answer*: a⁴/b *Concept*: Laws of division for exponents, negative exponents in simplified form. **Question 5**: Given (4/5)^x = (5/4)^(2x-3), find x. *Solution*: Step 1: Rewrite (4/5)^x as (5/4)^(-x) Step 2: (5/4)^(-x) = (5/4)^(2x-3) Step 3: Equate exponents: –x = 2x – 3 Step 4: –3x = –3, so x = 1 *Answer*: x = 1 *Concept*: Inverting fractional bases, equating exponents. *Tip*: For 3-mark questions, always show each step clearly. Examiners award partial credit for method even if the final answer is wrong. Allocate 4–5 minutes per question.

Five-mark questions combine multiple laws, algebraic manipulation, and often require proof or extended justification. **Question 1**: Simplify: [(2⁻¹ + 3⁻¹ + 4⁻¹)⁻¹] + [(5⁻¹ + 6⁻¹)⁻¹] *Solution*: *Part A*: (2⁻¹ + 3⁻¹ + 4⁻¹)⁻¹ Step 1: 1/2 + 1/3 + 1/4 = 6/12 + 4/12 + 3/12 = 13/12 Step 2: (13/12)⁻¹ = 12/13 *Part B*: (5⁻¹ + 6⁻¹)⁻¹ Step 1: 1/5 + 1/6 = 6/30 + 5/30 = 11/30 Step 2: (11/30)⁻¹ = 30/11 *Part C*: 12/13 + 30/11 = (12 × 11 + 30 × 13)/(13 × 11) = (132 + 390)/143 = 522/143 *Answer*: 522/143 (or ≈ 3.65) *Concept*: Negative exponents, fraction operations, combining results. **Question 2**: Prove that: (a^(m-n) × a^(n-p) × a^(p-m)) = 1, where a ≠ 0. *Solution*: Step 1: Apply the law a^x × a^y = a^(x+y) a^(m-n) × a^(n-p) × a^(p-m) = a^[(m-n)+(n-p)+(p-m)] Step 2: Expand the exponent: (m – n) + (n – p) + (p – m) = m – n + n – p + p – m = 0 Step 3: So the expression becomes a⁰ = 1 Step 4: Hence proved. *Concept*: Laws of exponents, proving identities, careful algebraic manipulation. **Question 3**: The mass of Earth is approximately 5.97 × 10²⁴ kg. The mass of an electron is approximately 9.11 × 10⁻³¹ kg. How many times heavier is Earth than an electron? Express your answer in scientific notation. *Solution*: Step 1: Ratio = (Mass of Earth) / (Mass of electron) = (5.97 × 10²⁴) / (9.11 × 10⁻³¹) Step 2: = (5.97 / 9.11) × (10²⁴ / 10⁻³¹) Step 3: = 0.655 × 10^(24-(-31)) = 0.655 × 10⁵⁵ Step 4: Convert to proper scientific notation: 6.55 × 10⁵⁴ *Answer*: 6.55 × 10⁵⁴ times *Concept*: Scientific notation, division of powers, real-world application. *Tip*: Problems involving Earth, atoms, and microscopic objects often use scientific notation. Practice converting between standard and scientific form without a calculator. *Tip*: Five-mark questions test your ability to integrate multiple concepts. Spend 7–8 minutes and prioritize showing working. Partial marks are common even if the final numerical answer is slightly off due to rounding.

Based on recent CBSE announcements and 2024–25 pilot papers, Chapter 10 will see the following emphasis changes: **1. Increased Case-Study Questions**: Expect one 4-mark integrated question involving real-world scenarios (e.g., 'A smartphone battery drains to 2⁻ᵗ of its original charge after t hours. When does it fall below 10% of capacity?'). These blend algebra with problem-solving. You will need to set up equations and solve using exponent rules, not just simplify given expressions. **2. More Negative and Fractional Exponents**: The 2024–25 sample papers show a 40% increase in questions requiring you to simplify expressions like (x^(1/2))^(3/4) or (27)^(-2/3). Mastering rational exponents and their conversion to roots (e.g., x^(1/2) = √x, x^(1/3) = ∛x) is now critical. **3. Fewer Rote 'Simplify' Questions**: CBSE is moving away from purely mechanical simplifications (e.g., 'Simplify 2⁵ × 2³'). Instead, expect 'Simplify and hence find the value if x = 3' or 'Prove that…' variants. **4. Scientific Notation in Context**: Expressing large or small numbers will increasingly appear within multi-part questions, not as standalone items. For example: 'The radius of an atom is ~10⁻¹⁰ m. How many atoms fit along a 1 mm line? Express your answer in scientific notation.' **5. Emphasis on Proof and Justification**: Short-answer questions (1–2 marks) now regularly ask 'Why is a⁰ = 1?' or 'Justify that (a/b)^n = a^n / b^n'. Rote memorization of laws is insufficient; you must understand and articulate the reasoning. **Action Plan**: Begin incorporating these patterns now. When solving past papers, pause after simplifying and ask: 'How would I explain why this works?' Attempt to write two-sentence justifications for each law before moving forward. This habit-building will earn bonus marks in 2026–27.

During your 90-minute exam, this is how to tackle Chapter 10 questions efficiently: **Before the Exam** (30 minutes prep the day before): • Write out all five laws on a card: product rule, quotient rule, power of a power, power of a product, zero exponent. • Memorize that a^(-n) = 1/(a^n), a^(1/n) = ⁿ√a, and a⁰ = 1 (for a ≠ 0). • Do 3–4 quick 1-mark questions to activate recall. **In the Exam** (Time allocation): 1. **Scan the Paper** (1 minute): Identify all Chapter 10 questions. Aim for 20–25 marks from this chapter alone (typical paper structure). 2. **Solve 1-Mark Questions First** (5 minutes for 5 questions): These are quick wins. Do them under 1 minute each. Never skip them. 3. **Tackle 3-Mark Questions** (15 minutes for 5 questions): Allocate 3 minutes per question. Write each step. If stuck, move on; come back later. 4. **Attack 5-Mark Questions** (20 minutes for 3–4 questions): Spend time reading the question carefully. Identify what it's testing (proof, simplification, or application). Work step-by-step. 5. **Review and Recalculate** (remaining time): Check your arithmetic, especially in fraction addition and exponent arithmetic. **Common Pitfalls to Avoid**: • **Forgetting negative signs**: (−2)² = 4, but −2² = −4. Be precise with brackets. • **Mixing up laws**: a^m × a^n = a^(m+n), but a^m + a^n ≠ a^(m+n). Addition and multiplication are different. • **Misplacing decimals in scientific notation**: 0.047 = 4.7 × 10⁻² (NOT 10²). Count decimal shifts carefully. • **Leaving negative exponents unsimplified**: Examiners expect 1/a² or a⁻², not 2⁻³ × 5. Simplify fully. • **Rushing through fraction inversions**: Double-check that (a/b)⁻¹ = (b/a)¹ = b/a. **Confidence Boosters**: • Solve at least 10 previous year questions before the exam. • Practice mental arithmetic: 2⁵, 3⁴, 5³, 10² should be instant recall. • Explain one law aloud to a friend or parent each day. This cements understanding. With this strategy, you can comfortably score 18–20 marks on Chapter 10 alone.

Use this checklist to verify your readiness before your exam: ☐ **Laws of Exponents**: I can apply all five laws (product, quotient, power of a power, power of a product, zero exponent) without referring to notes. ☐ **Negative Exponents**: I understand that a^(-n) means 1/(a^n) and can convert freely between a⁻³ and 1/a³. ☐ **Fractional Exponents**: I know that a^(1/2) = √a, a^(1/3) = ∛a, and a^(m/n) = ⁿ√(a^m). I can simplify (8)^(2/3) = (∛8)² = 2² = 4. ☐ **Scientific Notation**: I can express 5,00,000 as 5 × 10⁵ and 0.00045 as 4.5 × 10⁻⁴ without hesitation. ☐ **Simplification Steps**: I show all working, never skip steps, and always verify my answer by substituting a simple value (e.g., x = 2). ☐ **Proof Questions**: I can prove that a^m × a^n = a^(m+n) using the definition of exponents (repeated multiplication). ☐ **Fraction Handling**: I am confident adding, subtracting, and inverting fractions, especially within exponent problems like (2⁻¹ + 3⁻¹)⁻¹. ☐ **Speed**: I can solve a 1-mark question in under 1 minute, a 3-mark in 3–4 minutes, and a 5-mark in 6–8 minutes. ☐ **Common Errors**: I avoid mixing exponent rules (e.g., a^m × a^n ≠ (ab)^(m+n)) and am careful with negative signs. If you tick 8/8, you are exam-ready. If you miss any, revisit that topic using the solutions provided above.