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Exponents look simple until a test mixes them: a negative power here, a zero power there, a number with eleven digits to write in standard form. This post gives you the rules exactly as NCERT's new Class 8 book states them, six worked examples, the slips that cost the most marks, and a practice routine that sticks.
Where are exponents in the new Class 8 NCERT book?
In the new textbook, Ganita Prakash Part I (Reprint 2026-27), exponents are Chapter 2, "Power Play". In the older NCERT book they were Chapter 10, Exponents and Powers, as its contents page shows. Chapter 1 of the new book, "A Square and A Cube", also uses powers, so the two chapters support each other.
We write n^a for "n to the power a", meaning n multiplied by itself a times. The Power Play chapter lists these rules in its summary:
n^a × n^b = n^(a + b)
(n^a)^b = n^(a × b)
n^a ÷ n^b = n^(a − b), for n ≠ 0
n^a × m^a = (n × m)^a
n^a ÷ m^a = (n ÷ m)^a, for m ≠ 0
n^0 = 1, for n ≠ 0
n^(−a) = 1 ÷ n^a
Why is any number to the power 0 equal to 1?
The book uses the division rule. 2⁴ ÷ 2⁴ is a number divided by itself, which is 1. The rule says it is also 2^(4 − 4) = 2⁰. So 2⁰ must be 1 if the rule is to keep working.
Negative powers come the same way. Halve a 16-unit line five times: 2⁴ ÷ 2⁵ = 2⁻¹, and the line is ½ unit long. So 2⁻¹ = ½, and in general a negative power means "one over" the positive power.
Worked example 1. Simplify (2⁵ × 2⁻³) ÷ 2⁴.
Add, then subtract the exponents: 5 + (−3) − 4 = −2
Answer: 2⁻² = 1/4
Check with numbers: 32 × 1/8 = 4, and 4 ÷ 16 = 1/4
Worked example 2. Simplify (3²)³ × 3⁻⁴.
(3²)³ = 3⁶
3⁶ × 3⁻⁴ = 3² = 9
Worked example 3. Find 5³ × 2³ without a calculator.
Same exponent, so multiply the bases: (5 × 2)³ = 10³ = 1000
How do you write a number in standard form?
The book defines scientific notation, also called standard form, as x × 10^y, where x is at least 1 and less than 10, and y is an integer. Its own example: 308100000 = 3.081 × 10⁸. It also makes a sharp point: the exponent usually matters more than the first digits. Writing Mumbai's population of about 2 crore as 2 × 10⁷, changing the 2 to a 3 adds half again, but changing the 7 to an 8 makes it ten times larger.
Worked example 4. Write 4,05,00,000 in standard form.
In international commas this is 40,500,000
Put the decimal point after the first digit: 4.05
Count the places it moved: 7
Answer: 4.05 × 10⁷
Worked example 5. Calculate (3 × 10⁴) × (5 × 10⁶) and give the answer in standard form.
Multiply the coefficients: 3 × 5 = 15
Add the exponents: 10^(4 + 6) = 10¹⁰
15 × 10¹⁰ is not standard form, because 15 is not less than 10
15 × 10¹⁰ = 1.5 × 10¹ × 10¹⁰ = 1.5 × 10¹¹
Worked example 6. Which is larger, 3.2 × 10⁸ or 9.9 × 10⁷?
Compare exponents first: 10⁸ is ten times 10⁷
3.2 × 10⁸ = 32 × 10⁷, which is more than 9.9 × 10⁷
Because y can be any integer, small numbers work the same way: 0.00025 = 2.5 × 10⁻⁴. The chapter's examples are mostly large numbers, but science books use small ones often.
Which exponent mistakes come up most often?
Multiplying base and exponent. 2³ is 2 × 2 × 2 = 8, not 6.
Combining different bases. 2³ × 3² is 8 × 9 = 72. It is not 6⁵, which is 7776.
Reading a negative exponent as a negative number. 2⁻³ = 1/8, not −8.
Thinking 7⁰ = 0. Any non-zero number to the power 0 is 1.
Losing the bracket. (−3)² = 9, but −3² means −(3²) = −9. The chapter asks you to think about the sign of (−1)⁵ and (−1)⁵⁶ for the same reason.
Stopping too early in standard form. 15 × 10¹⁰ is correct in value but not in form.
Miscounting zeros. With Indian commas, count the digits after the first one, not the commas.
How should you practise exponents?
Mixing problem types beats doing one type at a time. In a study by Rohrer, Dedrick and Stershic, 126 seventh-grade students practised the same problems over three months, either grouped by type or interleaved. The interleaved group scored higher on tests given one day and 30 days later. Mixing forces you to choose the rule, which is exactly what a test asks of you.
Day 1: Write the seven rules from memory and test each with small numbers.
Day 2: Do a shuffled set of eight: two law questions, two zero or negative powers, two standard-form conversions, two comparisons.
Day 3: Find five large or small numbers in your science textbook and write them in standard form.
Day 4: Make an error log. For every wrong answer, write which of the seven slips above it was.
Day 5: Repeat a shuffled set of eight with new numbers.
Powers also run through squares and cubes; our guide to squares and square roots in Class 8 works through that chapter. When an exponent question goes wrong and you cannot see why, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer.
Start now: write 30,00,000 and 0.0006 in standard form, then check that your coefficients are between 1 and 10.
