This MCQ module is based on: Negative Exponents & Scientific Notation
TOPIC 5 OF 22
Negative Exponents & Scientific Notation
🎓 Class 8
Mathematics
CBSE
Theory
Ch 2 — Power Play (Exponents and Powers)
⏱ ~30 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Negative Exponents & Scientific Notation
Targeting Class 8 level in Algebra, with Basic difficulty.
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Chapter 2: Power Play
Class 8 Mathematics — Ganita Prakash Part-I | Part 2 of 3: Negative Exponents & Scientific Notation
2.3 The Other Side of Powers
Law 4: Division of Powers (Same Base)
Imagine a line of length \(2^4 = 16\) units. Each time we "halve" it, we divide by 2:
Halving a line step by step
Halve once: \(2^4 \div 2 = 2^4 \div 2^1 = 2^{4-1} = 2^3 = 8\) units
Halve twice: \(2^4 \div 2^2 = 2^{4-2} = 2^2 = 4\) units
Halve three times: \(2^4 \div 2^3 = 2^{4-3} = 2^1 = 2\) units
Law 4 — Division of Powers (same base):
\[n^a \div n^b = n^{a-b} \quad (n \neq 0,\ a > b)\]Example: \(2^{100} \div 2^{25} = 2^{75}\)
When \(a = b\): The Zero Exponent
What happens when we divide \(2^4\) by itself?
Deriving n⁰ = 1
\(2^0 = 2^{4-4} = 2^4 \div 2^4 = \dfrac{2\times2\times2\times2}{2\times2\times2\times2} = 1\)
In general: \(x^a \div x^a = x^{a-a} = x^0 = 1\)
\[n^0 = 1 \quad \text{for any } n \neq 0\]
Examples: \(5^0 = 1\), \((-7)^0 = 1\), \((1000)^0 = 1\), \(a^0 = 1\) (a ≠ 0)
When \(a < b\): Negative Exponents
What if we halve a line of \(2^4\) units more than 4 times?
Deriving negative exponents
Halving 5 times: \(2^4 \div 2^5 = 2^{4-5} = 2^{-1}\)
But also: \(\dfrac{2\times2\times2\times2}{2\times2\times2\times2\times2} = \dfrac{1}{2}\)
∴ \(2^{-1} = \dfrac{1}{2}\)
Halving 10 times: \(2^4 \div 2^{10} = 2^{-6}\), and \(\dfrac{2^4}{2^{10}} = \dfrac{1}{2^6} = \dfrac{1}{64}\)
∴ \(2^{-6} = \dfrac{1}{64}\)
\[n^{-a} = \frac{1}{n^a} \qquad \text{and} \qquad n^a = \frac{1}{n^{-a}} \quad (n \neq 0)\]
Examples: \(10^{-3} = \dfrac{1}{10^3} = \dfrac{1}{1000}\) | \(7^{-2} = \dfrac{1}{49}\) | \((-5)^{-2} = \dfrac{1}{25}\)
Think About It
Can we write \(10^3 = \dfrac{1}{10^{-3}}\)? Let's check:
\(\dfrac{1}{10^{-3}} = \dfrac{1}{\frac{1}{10^3}} = 1 \div \dfrac{1}{10^3} = 1 \times 10^3 = 10^3\) ✓
So \(7^2 = \dfrac{1}{7^{-2}}\) and \(4^a = \dfrac{1}{4^{-a}}\). Negative exponents are just reciprocals!
Power Lines — Visualising All Powers
Arranging all integer powers of a number on a vertical line makes the patterns crystal clear. Going up = multiply by the base. Going down = divide by the base.
Powers of 4
4⁸65,536
4⁷16,384
4⁶4,096
4⁵1,024
4⁴256
4³64
4²16
4¹4
4⁰1
4⁻¹1/4
4⁻²1/16
↑ ×4 each step ↑
↓ ÷4 each step ↓
↓ ÷4 each step ↓
Powers of 7
7⁷8,23,543
7⁶1,17,649
7⁵16,807
7⁴2,401
7³343
7²49
7¹7
7⁰1
7⁻¹1/7
7⁻²1/49
7⁻³1/343
7⁻⁴1/2,401
↑ ×7 each step ↑
↓ ÷7 each step ↓
↓ ÷7 each step ↓
Using Power Lines to compute
How many times larger is \(4^7\) than \(4^5\)?
\(4^7 \div 4^5 = 4^{7-5} = 4^2 = 16\). So \(4^7 = 16 \times 4^5\). ✓
How many times larger is \(4^2\) than \(4^{-2}\)?
\(4^2 \div 4^{-2} = 4^{2-(-2)} = 4^4 = 256\).
Power Line Exercises — Powers of 7
Use the power line above to find:
| Expression | Using Law | Answer |
|---|---|---|
| \(2401 \times 49\) | \(7^4 \times 7^2 = 7^6\) | 1,17,649 |
| \(49^3\) | \((7^2)^3 = 7^6\) | 1,17,649 |
| \(343 \times 2401\) | \(7^3 \times 7^4 = 7^7\) | 8,23,543 |
| \(16807 \div 49\) | \(7^5 \div 7^2 = 7^3\) | 343 |
| \(7 \div 343\) | \(7^1 \div 7^3 = 7^{-2}\) | 1/49 |
| \(16807 \div 823543\) | \(7^5 \div 7^7 = 7^{-2}\) | 1/49 |
| \(117649 \times \frac{1}{343}\) | \(7^6 \times 7^{-3} = 7^3\) | 343 |
| \(\frac{1}{343} \times \frac{1}{343}\) | \(7^{-3} \times 7^{-3} = 7^{-6}\) | 1/1,17,649 |
Figure it Out: Simplify using laws of exponents
Write equivalent forms: (i) \(2^{-4}\) (ii) \(10^{-5}\) (iii) \((-7)^{-2}\) (iv) \((-5)^{-3}\) (v) \(10^{-100}\)
Simplify: (i) \(2^{-4} \times 2^7\) (ii) \(3^2 \times 3^{-5} \times 3^6\) (iii) \(p^3 \times p^{-10}\) (iv) \(2^4 \times (-4)^{-2}\) (v) \(8^p \times 8^q\)
Equivalent forms:
(i) \(2^{-4} = \frac{1}{16}\) (ii) \(10^{-5} = \frac{1}{10^5}\) (iii) \((-7)^{-2} = \frac{1}{49}\) (iv) \((-5)^{-3} = \frac{-1}{125}\) (v) \(10^{-100} = \frac{1}{10^{100}}\)
Simplify:
(i) \(2^{-4+7} = 2^3 = 8\)
(ii) \(3^{2-5+6} = 3^3 = 27\)
(iii) \(p^{3-10} = p^{-7} = \frac{1}{p^7}\)
(iv) \((-4)^{-2} = \frac{1}{16}\), so \(2^4 \times \frac{1}{16} = 16 \times \frac{1}{16} = 1 = 2^0\)
(v) \(8^{p+q}\)
2.4 Powers of 10 — Expanded Form
We use powers of 10 when writing numbers in expanded form. For a whole number:
Expanded form using powers of 10
\(47561 = (4 \times 10^4) + (7 \times 10^3) + (5 \times 10^2) + (6 \times 10^1) + (1 \times 10^0)\)
For decimals, we use negative exponents:
\(561.903 = (5 \times 10^2) + (6 \times 10^1) + (1 \times 10^0) + (9 \times 10^{-1}) + (0 \times 10^{-2}) + (3 \times 10^{-3})\)
Negative powers of 10 represent decimal place values
Scientific Notation (Standard Form)
Very large (or very small) numbers are hard to read correctly — one misplaced zero can mean the difference between ₹5,000 and ₹50,000!
solves this by writing every number as \(x \times 10^y\), where \(1 \leq x < 10\) and \(y\) is any integer.
Converting to Scientific Notation
5900 = 5.9 × 10³
20800 = 2.08 × 10⁴
80,00,000 = 8 × 10⁶
59,853 = 5.9853 × 10⁴
34,30,000 = 3.43 × 10⁶
70,04,00,00,000 = 7.004 × 10¹⁰
Move the decimal left until 1 ≤ coefficient < 10. Count moves = exponent.
Why the Exponent Matters More
When Mumbai's population is written as \(2 \times 10^7\):
- Changing the coefficient 2 → 3 means a 50% increase (2 crore to 3 crore)
- Changing the exponent 7 → 8 means a 10× increase (2 crore to 20 crore)
The exponent indicates the order of magnitude — how many digits the number has. It is more important than the coefficient!
Real-world astronomical distances
Sun to Milky Way centre: \(3 \times 10^{20}\) m
Number of stars in our galaxy: \(1 \times 10^{11}\)
Mass of Earth: \(5.976 \times 10^{24}\) kg
Sun–Saturn distance: \(1.4335 \times 10^{12}\) m
Sun–Earth distance: \(1.496 \times 10^{11}\) m
Sun–Uranus distance: \(1.439 \times 10^{12}\) m
The Sun–Saturn distance is ~10 times the Sun–Earth distance — immediately visible from the exponents (10¹² vs 10¹¹)
Linear Growth vs. Exponential Growth
Linear (Additive) Growth
Increases by a fixed amount each step
20 + 20 + 20 + ...
(1,92,20,00,000 times)
Steps to Moon at 20 cm/step:
≈ 192 crore steps!
Exponential (Multiplicative) Growth
Multiplied by a fixed factor each step
0.001 × 2 × 2 × ...
(just 46 times)
Paper folds to Moon:
Only 46 folds!
Exponential growth appears slow initially but outpaces linear growth dramatically
Key insight: Linear growth is additive (add the same amount repeatedly). Exponential growth is multiplicative (multiply by the same factor repeatedly). The Moon-ladder example needs 192 crore steps; but paper folding reaches there in just 46 folds. That's the power of exponential growth!
Interactive: Scientific Notation Converter
Scientific Notation Converter
All Laws of Exponents — Summary
Complete Laws of Exponents
| Law | Formula | Example |
|---|---|---|
| Product (same base) | \(n^a \times n^b = n^{a+b}\) | \(2^3 \times 2^5 = 2^8\) |
| Power of power | \((n^a)^b = n^{ab}\) | \((3^2)^4 = 3^8\) |
| Division (same base) | \(n^a \div n^b = n^{a-b}\) | \(5^7 \div 5^3 = 5^4\) |
| Same exponent (product) | \(m^a \times n^a = (mn)^a\) | \(2^4 \times 3^4 = 6^4\) |
| Same exponent (quotient) | \(m^a \div n^a = (m/n)^a\) | \(10^4 \div 5^4 = 2^4\) |
| Zero exponent | \(n^0 = 1\) (n≠0) | \(7^0 = 1\) |
| Negative exponent | \(n^{-a} = 1/n^a\) | \(2^{-3} = 1/8\) |
Competency-Based Questions
Q1. Write \(\frac{1}{10000}\) in exponential form using (i) positive exponent, (ii) negative exponent. L1 Remember
(i) \(\frac{1}{10000} = \frac{1}{10^4} = 10^{-4}\) — that's already with negative exponent.
(ii) With positive: \(\frac{1}{10^4}\).
Both represent the same value. The negative exponent notation \(10^{-4}\) is more compact.
Q2. The distance from Earth to the nearest star (Proxima Centauri) is approximately \(4 \times 10^{16}\) m. The distance from Earth to the Sun is \(1.5 \times 10^{11}\) m. How many times farther is Proxima Centauri than the Sun? Express your answer as a power of 10. L3 Apply
\(\frac{4 \times 10^{16}}{1.5 \times 10^{11}} = \frac{4}{1.5} \times 10^{16-11} = 2.67 \times 10^5 \approx 2.7 \times 10^5\)
Proxima Centauri is about 2,70,000 times farther than the Sun!
Q3. Simplify and write in exponential form: \(\dfrac{(3^4)^2 \times 3^{-5}}{3^3}\) L3 Apply
\((3^4)^2 = 3^8\)
Numerator: \(3^8 \times 3^{-5} = 3^{8+(-5)} = 3^3\)
Full expression: \(3^3 \div 3^3 = 3^0 = \mathbf{1}\)
Q4. Nandita says "10⁻⁵ is smaller than 10⁻³ because 5 > 3." Deepak says "No, 10⁻⁵ is smaller, but the reason is \(10^{-5} = \frac{1}{10^5}\) and \(\frac{1}{10^5} < \frac{1}{10^3}\)." Who gave the correct reasoning? L4 Analyse
Deepak gave the correct reasoning. Nandita's statement (10⁻⁵ is smaller) is correct, but her reasoning is incomplete/misleading. The correct explanation is: \(10^{-5} = \frac{1}{10^5} = 0.00001\) and \(10^{-3} = \frac{1}{10^3} = 0.001\), and \(0.00001 < 0.001\). ✓
Q5. A bacteria colony doubles every hour. At noon, there are \(10^6\) bacteria. Write expressions for the count at (i) 3 PM, (ii) 9 AM. Express in scientific notation. Comment on which direction (past or future) requires negative exponents. L5 Evaluate
(i) 3 PM = noon + 3 hours: \(10^6 \times 2^3 = 10^6 \times 8 = 8 \times 10^6\)
(ii) 9 AM = noon − 3 hours: \(10^6 \times 2^{-3} = 10^6 \times \frac{1}{8} = \frac{10^6}{8} = 1.25 \times 10^5\)
Going backward in time (before noon) requires negative exponents because the colony was smaller — halving with each hour backward. Negative exponents capture "undoing" multiplication, i.e., division.
Assertion–Reason Questions
Options: (A) Both true, Reason is correct explanation. (B) Both true, Reason is NOT correct explanation. (C) Assertion true, Reason false. (D) Assertion false, Reason true.
ARQ 1.
Assertion: \(n^0 = 1\) for all values of \(n\).
Reason: \(n^a \div n^a = n^{a-a} = n^0\), and any non-zero number divided by itself equals 1.
(C) — The Assertion is partially false: \(n^0 = 1\) only for \(n \neq 0\). The expression \(0^0\) is undefined. The Reason correctly explains why \(n^0 = 1\) for non-zero \(n\), but since the Assertion doesn't specify \(n \neq 0\), it is not fully correct.
ARQ 2.
Assertion: \(3.08 \times 10^8\) represents a number with 9 digits.
Reason: In scientific notation \(x \times 10^y\), the exponent \(y\) indicates the number of digits is \(y+1\) (when \(x\) has one digit before decimal).
(A) — Both true. \(3.08 \times 10^8 = 30{,}80{,}00{,}000\), which has 9 digits. The Reason correctly explains: exponent 8 means \(8+1 = 9\) digits. ✓
ARQ 3.
Assertion: Exponential growth is always faster than linear growth.
Reason: In exponential growth, each step multiplies the quantity, while in linear growth each step adds a fixed amount; multiplication produces larger numbers faster than addition.
(A) — Both true. For any fixed base > 1 and any fixed addition amount, exponential growth will eventually outpace linear growth (and sustain that lead). The paper folding vs Moon-ladder comparison is a perfect illustration.
Frequently Asked Questions — Power Play (Exponents and Powers)
What is Negative Exponents & Scientific Notation in NCERT Class 8 Mathematics?
Negative Exponents & Scientific Notation is a key concept covered in NCERT Class 8 Mathematics, Chapter 2: Power Play (Exponents and Powers). This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.
How do I solve problems on Negative Exponents & Scientific Notation step by step?
To solve problems on Negative Exponents & Scientific Notation, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 8 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.
What are the most important formulas for Chapter 2: Power Play (Exponents and Powers)?
The essential formulas of Chapter 2 (Power Play (Exponents and Powers)) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.
Is Negative Exponents & Scientific Notation important for the Class 8 board exam?
Negative Exponents & Scientific Notation is part of the NCERT Class 8 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.
What mistakes should students avoid in Negative Exponents & Scientific Notation?
Common mistakes in Negative Exponents & Scientific Notation include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.
Where can I find more NCERT practice questions on Negative Exponents & Scientific Notation?
End-of-chapter NCERT exercises for Negative Exponents & Scientific Notation cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 2, and solve at least one previous-year board paper to consolidate your understanding.
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