Looking for the Exponents and Powers Class 8 chapter? This page gives you NCERT Class 8 Mathematics Chapter 10 — the chapter on negative exponents, the six laws of exponents, and writing very small numbers in standard form. It runs from page 121 to page 129 of the NCERT book, and the official chapter PDF is right below.
Download the NCERT Class 8 Exponents and Powers Chapter 10 PDF
If you need the chapter exactly as printed — every rule, worked example and both exercises — the official NCERT Class 8 Mathematics Chapter 10 Exponents and Powers PDF is published by NCERT on its own website, so you can open it in your browser or save the file to study offline.
Chapter 10 at a Glance: What the PDF Contains
This table lists what the chapter holds before you open the file — its sections, figures, worked examples and exercise questions — so you can see at a glance how much there is.
What NCERT Chapter 10 Covers, Section by Section
This is the order the chapter actually follows. If you are stuck on one idea, jump to its row and open that page.
| Section | What it teaches | NCERT pages |
|---|---|---|
| 10.1 Introduction | Why exponents exist: the Earth’s mass, \(5,970,000,000,000,000,000,000,000\) kg, written compactly as \(5.97 \times 10^{24}\) kg, read as “10 raised to the power 24” | p. 121 |
| 10.2 Powers with Negative Exponents | The divide-by-the-base pattern that gives \(10^{-1} = \frac{1}{10}\), \(10^{-2} = \frac{1}{100}\), and the general rule \(a^{-m} = \frac{1}{a^m}\) | pp. 122–123 |
| 10.3 Laws of Exponents | The six laws of exponents, verified for negative exponents, with worked examples including the method for finding an unknown exponent | pp. 124–125 |
| Exercise 10.1 | Practises evaluation and simplification with negative exponents, ending with compound fractions | p. 126 |
| 10.4 Use of Exponents to Express Small Numbers in Standard Form | Writing very small numbers in standard form — the chapter opens with real facts like the red blood cell \(0.000007\) mm, paper \(0.0016\) cm and Mount Everest \(8848\) m | pp. 127–128 |
| 10.4.1 Comparing very large and very small numbers | Comparing sizes by dividing standard forms — Sun vs Earth, red blood cells vs plant cells — and adding numbers after matching exponents | p. 128 |
| Exercise 10.2 | Practises standard form and usual form, plus one word problem on the total thickness of a stack of books | p. 129 |
| What Have We Discussed? | The chapter’s closing recap: negative exponents obey the same laws; very small numbers go into standard form | p. 129 |
How Negative Exponents Work: The Pattern That Defines Them
Students often read \(10^{-1}\) as “minus ten”. That is wrong. A negative exponent is shorthand for one over the positive power — a reciprocal — and the chapter builds the idea from a pattern you already know.
Write the powers of 10 in order and watch what happens at each step:
| Power | How it is built | Value |
|---|---|---|
| \(10^3\) | \(10 \times 10 \times 10\) | 1000 |
| \(10^2\) | \(1000 \div 10\) | 100 |
| \(10^1\) | \(100 \div 10\) | 10 |
| \(10^0\) | \(10 \div 10\) | 1 |
| \(10^{-1}\) | \(1 \div 10\) | \(\frac{1}{10}\) |
| \(10^{-2}\) | \(\frac{1}{10} \div 10\) | \(\frac{1}{100}\) |
| \(10^{-3}\) | \(\frac{1}{100} \div 10\) | \(\frac{1}{1000}\) |
Every step divides by the base, 10. As the exponent decreases by 1, the value becomes one-tenth of the previous value (NCERT, p. 122). The same pattern with base 3 gives \(3^2 = 9\) because \(27 \div 3 = 9\), and finally \(3^{-3} = \frac{1}{27}\).
Written as a rule: for any non-zero integer \(a\), \(a^{-m} = \frac{1}{a^m}\), where \(m\) is a positive integer. So \(a^{-m}\) is the multiplicative inverse of \(a^m\) (NCERT, p. 122) — the two multiply to give 1. The base cannot be zero because dividing by zero is undefined.
This is also how decimals appear in expanded form. The book writes \[ 1425.36 = 1 \times 10^3 + 4 \times 10^2 + 2 \times 10 + 5 \times 1 + 3 \times 10^{-1} + 6 \times 10^{-2} \]
Tenths are \(10^{-1}\) and hundredths are \(10^{-2}\) — a negative exponent is simply the link between powers and decimal places (NCERT, p. 123).
The Six Laws of Exponents, Now Valid for All Integers
You learned these six laws in Class VII for positive exponents only. Section 10.3 verifies that the same laws hold when the exponents are negative integers — that verification is what lets you apply them everywhere (NCERT, p. 124).
| Law | Rule | In words |
|---|---|---|
| Product of powers | \(a^m \times a^n = a^{m+n}\) | Same base? Add the exponents. |
| Quotient of powers | \(a^m \div a^n = a^{m-n}\) | Same base? Subtract the exponents. |
| Power of a power | \((a^m)^n = a^{mn}\) | Multiply the exponents. |
| Power of a product | \(a^m \times b^m = (ab)^m\) | The exponent belongs to every factor. |
| Power of a quotient | \(a^m \div b^m = \left(\frac{a}{b}\right)^m\) | The exponent belongs to numerator and denominator. |
| Zero exponent | \(a^0 = 1\) | Any non-zero base raised to the power zero is 1. |
For these laws, \(a\) and \(b\) are non-zero integers and \(m, n\) are any integers — positive, negative or zero (NCERT, p. 124).
Here is one worked example that uses three of the laws together, with fresh numbers.
Problem: Simplify \(\dfrac{5^{-2} \times 5^{7} \times 5^{0}}{5^{3}}\) and write the answer as a whole number.
Step 1: Use the product law and the zero-exponent law in the numerator: \(5^{-2} \times 5^{7} \times 5^{0} = 5^{-2+7+0} = 5^{5}\).
\[ \frac{5^{-2} \times 5^{7} \times 5^{0}}{5^{3}} = \frac{5^{5}}{5^{3}} \]
Step 2: Use the quotient law: \(5^{5} \div 5^{3} = 5^{5-3} = 5^{2}\).
\[ 5^{5-3} = 5^{2} = 25 \]
Final answer: \(5^{2} = 25\).
One more idea from the same pages decides many “find the value of \(m\)” questions. If \((-3)^{m+1} \times (-3)^5 = (-3)^7\), add the exponents on the left: \((-3)^{m+6} = (-3)^7\).
Because the bases are the same and that base is different from 1 and \(-1\), the exponents must be equal, so \(m + 6 = 7\) and \(m = 1\) (NCERT, p. 125).
Why the restriction? NCERT states it plainly: \(a^n = 1\) only if \(n = 0\) for a general base, but \(1^n = 1\) for infinitely many \(n\), and \((-1)^p = 1\) for every even integer \(p\) (NCERT, p. 125).
If the base were 1 or \(-1\), two equal powers would not force equal exponents — so the comparison step is only safe when the base is neither.
Reading the Chapter’s Key Figures, One by One
The figures in this chapter are equation chains, not pictures. Each one shows a step the text is making, so learning to read them is learning the topic.
This chain repeats the opening pattern with a different base: \(3^3 = 27\), then \(27 \div 3 = 9 = 3^2\), down to \(3^{-3} = \frac{1}{27}\). Read it downward and you see exactly why a negative exponent is a fraction — each step divides the previous value by the base (NCERT, p. 122).
This boxed rule is the chapter’s core: a negative exponent on any non-zero integer \(a\) means the reciprocal of \(a^m\). Since \(a^{-m} \times a^m = \frac{1}{a^m} \times a^m = 1\), the two powers are multiplicative inverses — one undoes the other (NCERT, p. 122).
This figure opens the worked examples of section 10.3. Its first lines show \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\) and \(\frac{1}{3^{-2}} = 3^2 = 9\) — notice that a negative exponent in a denominator becomes positive when it moves to the numerator (NCERT, p. 124).
This is the decisive line of Example 5. After adding exponents, the equation reads \((-3)^{m+6} = (-3)^7\). The caption states the reasoning: same base, different from 1 and \(-1\), so the exponents must be equal — giving \(m = 1\). That is the exact sentence you should write in your answer (NCERT, p. 125).
This figure shows the chapter’s method for standard form on paper thickness: \(0.0016 = \frac{16}{10000} = 1.6 \times 10^{-3}\) cm (NCERT, p. 127).
The two-step method hiding inside it: step 1, move the decimal point so exactly one non-zero digit is before it — \(0.0016 \to 1.6\), three places right. Step 2, take that count as the exponent of 10, negative because you moved right: \(1.6 \times 10^{-3}\). Move left and the exponent is positive.
Definitions: Exponent, Base, Standard Form, Usual Form
These are the terms you must write correctly in answers. Each definition is one line.
| Term | Definition | NCERT page |
|---|---|---|
| Exponent | The raised number that counts how many times the base is used as a factor; “10 raised to the power 24” has exponent 24 | p. 121 |
| Base | The number being raised — the factor that repeats | p. 121 |
| Negative exponent | Shorthand for the reciprocal: \(a^{-m} = \frac{1}{a^m}\); it never means a negative number | p. 122 |
| Multiplicative inverse | The pair that multiplies to 1: \(a^{-m}\) is the multiplicative inverse of \(a^m\) | p. 122 |
| Standard form | A number written as \(k \times 10^n\) with \(1 \le k \lt 10\), e.g. \(7 \times 10^{-6}\) m and \(1.6 \times 10^{-3}\) cm | p. 127 |
| Usual form | Standard form expanded back into an ordinary number, e.g. \(3.52 \times 10^5 = 352000\) | p. 128 |
Common Mistakes in Exponents and How to Fix Them
These are the errors the chapter’s own cautions and examples warn about. Name them before you make them.
| Mistake | Correct rule | How to check your answer |
|---|---|---|
| Reading \(3^{-2}\) as \(-9\) | A negative exponent means the reciprocal: \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\) | Multiply your answer by \(3^2\). If the product is 1, the answer is right. |
| Concluding \(a^n = 1\) forces \(n = 0\) for every base | True only for a general base. \(1^n = 1\) for any \(n\), and \((-1)^p = 1\) for any even \(p\) | Name the base before comparing exponents. The “same base \(\Rightarrow\) equal exponents” step is invalid for 1 and \(-1\) (NCERT, p. 125). |
| Adding standard-form numbers without matching exponents | Convert both numbers to the same power of 10 first: \(5.97 \times 10^{24} = 597 \times 10^{22}\), then add | After adding, exactly one non-zero digit must sit before the decimal point (NCERT, p. 128). |
| Writing \(0.0016\) as \(16 \times 10^{-4}\) | Standard form needs \(1 \le k \lt 10\), so \(0.0016 = 1.6 \times 10^{-3}\) | Count the places the decimal point moved; the exponent equals that count, with the sign set by direction (NCERT, p. 127). |
| Dropping the sign when adding exponents on a negative base | Add exponents with their signs: \((-4)^5 \times (-4)^{-10} = (-4)^{5 + (-10)} = (-4)^{-5} = \frac{1}{(-4)^5}\) | The new exponent must equal the algebraic sum of the two exponents (NCERT, p. 124). |
Exponents and Powers Class 8: What Exercise 10.1 and Exercise 10.2 Test
Practice is split into two exercises. Each tests a different skill, so revise the one that matches where you are stuck.
| Exercise | Questions | Skill it tests |
|---|---|---|
| Exercise 10.1 (NCERT, p. 126) | Q1 | Evaluating negative powers directly: \(3^{-2}\), \((-4)^{-2}\), \(\left(\frac{1}{2}\right)^{-5}\) |
| Q2 | Simplifying and expressing the result in power notation with positive exponent | |
| Q3, Q4, Q6 | Combining several laws — zero exponent, power of a power, powers of fractions | |
| Q5 | Finding the unknown exponent \(m\): \(5^m \div 5^{-3} = 5^5\) | |
| Q7 | Simplifying compound fractions such as \(\frac{25 \times t^{-4}}{5^{-3} \times 10 \times t^{-8}}\) | |
| Exercise 10.2 (NCERT, p. 129) | Q1, Q3 | Expressing numbers and real-life facts in standard form |
| Q2 | Expressing standard form back in usual form | |
| Q4 | Word problem: total thickness of a stack of books and paper sheets |
The question shapes repeat across practice sets: “evaluate”, “simplify and express with positive exponents”, “find the value of \(m\)”, “express in standard form”, “express in usual form”. Recognise the shape and you know which law to reach for.
For full marks, a “simplify and express” answer must end with no negative exponent — write the reciprocal form if one remains. A “find the value of \(m\)” answer states the same-base reasoning (“same base, not 1 or \(-1\), so exponents are equal”) before solving for \(m\). A standard-form answer keeps exactly one non-zero digit before the decimal point.
One caution: textbook contents and the examinable syllabus are not always identical. Check the current official syllabus at cbse.gov.in before deciding what to practise.
The Chapter’s Closing Points, Explained Simply
The chapter ends on p. 129 with a closing section called “What Have We Discussed?”, and it reduces the whole topic to two points.
First, numbers with negative exponents obey the same six laws of exponents as numbers with positive exponents. Second, very small numbers can be expressed in standard form using negative exponents. If you can explain both sentences and apply the six laws, you have the chapter.
Related Resources for Class 8 Maths
This listing is maintained for the 2026-27 academic session using the NCERT textbook information available to us. NCERT remains the authority for confirming the latest edition.
Exponents and Powers sits between two other chapters in the NCERT Class 8 Mathematics book: Chapter 9 Mensuration comes before it, and Chapter 11 Direct and Inverse Proportions follows it. For the full set of chapter pages, use the Class 8 Mathematics page inside the Class 8 hub; the index of every subject sits under CBSE Notes.
The official NCERT website at ncert.nic.in hosts the current editions of all textbook files.
| What the chapter holds | Count | Where it is used |
|---|---|---|
| Printed pages | 8 | |
| Sections in the chapter | 5 | |
| Figures with NCERT captions | 5 | |
| Tables | 1 | |
| Worked examples | 5 | solved step by step in our NCERT Solutions |
| Exercise questions | 11 | answered in our NCERT Solutions |
| In-text questions | 2 | |
| Official NCERT PDF | Download the chapter PDF | the chapter exactly as NCERT publishes it |
Reference: NCERT Class 8 Mathematics textbook, chapter 10, official edition on ncert.nic.in.
Sources and data verification
- Chapter data and figures on this page describe the NCERT Class 8 Mathematics textbook, Chapter 10 Exponents and Powers, English edition, PDF published by NCERT.
- This page covers that one chapter only — not other chapters of the book and not other NCERT textbooks.
- It is maintained for the current academic session using the NCERT information available to us on ncert.nic.in.
- NCERT settles textbook text, editions and PDFs; CBSE settles the curriculum, syllabus and examinations.
Frequently Asked Questions
What is a negative exponent and how do you simplify it?
A negative exponent means the reciprocal, never a negative number: \(a^{-m} = \frac{1}{a^m}\), so \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\). To simplify, evaluate the positive power and take its reciprocal; a negative exponent in a denominator moves to the numerator as a positive one, so \(\frac{1}{3^{-2}} = 3^2 = 9\).
How do you write a very small number such as 0.000007 in standard form?
Move the decimal point to the right until one non-zero digit stands before it, and use the number of places moved as a negative exponent of 10. \(0.000007 \to 7\) takes six places, so \(0.000007 = 7 \times 10^{-6}\) (NCERT, p. 127). Paper thickness works the same way: \(0.0016 = 1.6 \times 10^{-3}\) cm.
Do the laws of exponents still work when the exponents are negative?
Yes. Section 10.3 checks each law for negative exponents — for instance \(2^{-3} \times 2^{-2} = 2^{-5}\) and \(5^{-2} \times 5^4 = 5^2\) — then states the laws hold for any integers \(m\) and \(n\) (NCERT, p. 124).
When does a^n equal 1 and why are bases 1 and -1 special cases?
For a general base, \(a^n = 1\) only when \(n = 0\). The exceptions are the bases 1 and \(-1\): \(1^n = 1\) for every \(n\), and \((-1)^p = 1\) for every even \(p\). That is why the chapter’s “same base \(\Rightarrow\) equal exponents” step in Example 5 demands a base different from 1 and \(-1\) (NCERT, p. 125).
How do you add or compare numbers written in standard form?
First rewrite every number with the same power of 10, then add or compare the front parts. The book adds the Earth’s mass \(5.97 \times 10^{24}\) kg to the Moon’s \(7.35 \times 10^{22}\) kg by converting the first to \(597 \times 10^{22}\) kg, giving \(604.35 \times 10^{22}\) kg (NCERT, p. 128).
Which exercise covers which skill: Exercise 10.1 or Exercise 10.2?
Exercise 10.1 practises evaluating and simplifying expressions with negative exponents — including “find the value of \(m\)” and compound fractions. Exercise 10.2 practises standard form and usual form, plus one word problem on the total thickness of a stack of books and paper sheets (NCERT, pp. 126 and 129).
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