MathsCBSE Class 8 · Power Play

Powers of 10 and Scientific Notation for Class 8

Q: Why do we write 5.9 x 10^3 and not 59 x 10^2?

Both equal 5900, but standard form fixes the coefficient between 1 and 10 so that numbers are easy to compare by their exponents.

Q: What is the difference between a billion and an arab?

Both are 10^9: a billion in the international system and an arab in the Indian system.

Q: What does the number of digits in the coefficient mean?

It shows how precisely the number is known: 1.42 x 10^5 claims more precision than 1.4 x 10^5.

Q: Can the exponent be zero?

Yes. A number between 1 and 10 is written with exponent 0, for example 4.7 = 4.7 x 10^0.

Updated 17 October 2026 · Fact-checked

Scientific notation (standard form) writes a number as x times 10^y, where 1 <= x < 10 and y is an integer. Expanded form uses powers of 10 for each place value, including negative powers after the decimal point. Class 8 exams ask for conversion in both directions, products of numbers in standard form, comparison of huge quantities, and estimation with powers of 10.

Move the decimal point until one non-zero digit is before it; count the moves: left gives a positive exponent, right gives a negative one.

Illustration drawn for QuestionBuddy.

Understand Powers of 10 and Scientific Notation

Place value with powers of 10. Every digit's place value is a power of 10: $47561 = 4 \times 1 0^{4} + 7 \times 1 0^{3} + 5 \times 1 0^{2} + 6 \times 1 0^{1} + 1 \times 1 0^{0}$ . After the decimal point the powers are negative: $561.903 = 5 \times 1 0^{2} + 6 \times 1 0^{1} + 1 \times 1 0^{0} + 9 \times 1 0^{- 1} + 0 \times 1 0^{- 2} + 3 \times 1 0^{- 3}$ .

Scientific (standard) form. Any number can be written as $x \times 1 0^{y}$ with $1 \leq x < 10$ (the coefficient) and an integer exponent $y$ :

$5900 = 5.9 \times 1 0^{3}$ , $20800 = 2.08 \times 1 0^{4}$ , $80, 00, 000 = 8 \times 1 0^{6}$ .
Numbers below 1 use a negative exponent: $0.0035 = 3.5 \times 1 0^{- 3}$ .

To convert, move the decimal point until exactly one non-zero digit stands before it. Count the places moved: moving left gives a positive exponent, moving right gives a negative one.

Why use it. In very large numbers the count of zeros matters more than the first digits, and it is easy to miscount zeros or misplace commas. The exponent tells you at once how many digits the number has. The digits of the coefficient show how precisely you know the number: $1.42 \times 1 0^{5}$ claims more accuracy than $1.4 \times 1 0^{5}$ . The exponent matters most; changing the coefficient from 2 to 3 changes $2 \times 1 0^{7}$ by one half, but changing the exponent from 7 to 8 makes it ten times larger.

Comparing. Compare exponents first, then coefficients. The NCERT chapter lists these distances: the Sun to Saturn is $1.4335 \times 1 0^{12}$ m, Saturn to Uranus is $1.439 \times 1 0^{12}$ m, and the Sun to Earth is $1.496 \times 1 0^{11}$ m. The Sun to Earth distance is the smallest because its exponent is 11, one less than the others.

Multiplying and dividing. Multiply the coefficients and add the exponents, then adjust to bring the coefficient into $[1, 10)$ : $(3 \times 1 0^{5}) \times (4.5 \times 1 0^{- 2}) = 13.5 \times 1 0^{3} = 1.35 \times 1 0^{4}$ .

Names of large numbers. Indian system: a lakh is $1 0^{5}$ , a crore $1 0^{7}$ , an arab $1 0^{9}$ , a kharab $1 0^{11}$ , a neel $1 0^{13}$ , a padma $1 0^{15}$ , a shankh $1 0^{17}$ and a maha shankh $1 0^{19}$ . International system: a million is $1 0^{6}$ , a billion $1 0^{9}$ , a trillion $1 0^{12}$ and a quadrillion $1 0^{15}$ . A googol is $1 0^{100}$ .

Linear versus exponential growth. Adding a fixed amount each step is linear growth; multiplying each step is exponential growth. In the chapter, a 20 cm step ladder to the Moon (384,400 km away) would need 1,92,20,00,000 steps, whereas folding a 0.001 cm sheet reaches the Moon in 46 folds.

Estimation. Many real questions have no exact data. Describe the relationships, make reasonable stated assumptions, and compute with powers of 10.

The exponent laws used here are on the previous topic page.

Powers of 10 and Scientific Notation: formulas to remember

Scientific notation: $x \times 1 0^{y}, 1 \leq x < 10, y \in Z$; The standard form of a number.
Product in standard form: $(a \times 1 0^{m}) (b \times 1 0^{n}) = ab \times 1 0^{m + n}$; Re-adjust if ab is 10 or more.
Quotient in standard form: $\frac{a \times 1 0 ^{m}}{b \times 1 0 ^{n}} = \frac{a}{b} \times 1 0^{m - n}$; Re-adjust if a/b is below 1.
Indian large-number names: $lakh 1 0^{5}, crore 1 0^{7}, arab 1 0^{9}$; Kharab 10^11, neel 10^13, padma 10^15.
Seconds in a 365-day year: $365 \times 24 \times 3600 = 31, 536, 000 \approx 3.15 \times 1 0^{7}$; A derived value for time estimates.

How to solve Powers of 10 and Scientific Notation questions

Use these steps for conversion, calculation and estimation in standard form.

1To convert a number to standard form, place the decimal point after the first non-zero digit and count the places you moved.
2Moving the point left gives a positive exponent; moving it right gives a negative one.
3For a product or quotient, handle coefficients and powers of 10 separately.
4If the new coefficient is not between 1 and 10, shift the point and change the exponent to match.
5To compare, compare exponents first, then coefficients.
6For estimation, state each assumption, compute with powers of 10, and say how close you expect the answer to be.

Powers of 10 and Scientific Notation in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Standard form for counts

Suppose a library holds 3,25,000 books and a city of 2.5 crore people has such libraries.

Write both numbers in standard form and say how many times larger the city population is than the number of books, to the nearest whole number.

Try it first, then show the solutionSolution

$3, 25, 000 = 3.25 \times 1 0^{5}$ and $2.5 crore = 2.5 \times 1 0^{7}$ .
Ratio $= \frac{2.5 \times 1 0 ^{7}}{3.25 \times 1 0 ^{5}} = \frac{2.5}{3.25} \times 1 0^{2} \approx 0.769 \times 100 \approx 77$ .

Answer: About 77 times larger.

Memory hook: Same units of 10: subtract exponents (7 - 5 = 2), then divide coefficients.

Time in standard form

Imagine a computer program that checks one code every second.

The program has to check 10^9 codes. About how many years will it take, using a 365-day year?

Try it first, then show the solutionSolution

Seconds in a year $= 365 \times 24 \times 3600 = 31, 536, 000 \approx 3.15 \times 1 0^{7}$ .
Years $= \frac{1 0 ^{9}}{3.15 \times 1 0 ^{7}} \approx 31.7$ .

Answer: About 31.7 years.

Memory hook: Convert time to the same unit (seconds), then divide powers of 10.

Tricks to remember Powers of 10 and Scientific Notation

Short memory aids. Say each one aloud once, then use it on a question the same day.

One digit before the point
The coefficient has exactly one non-zero digit in front of the decimal point. If it has two, shift again.
Use it when: Checking that an answer is truly in standard form.
Left is plus, right is minus
Moving the decimal left makes the exponent positive; moving it right makes it negative.
Use it when: Converting numbers above and below 1.
Exponent first, then coefficient
Compare the exponents first; only if they are equal compare the coefficients.
Use it when: Ordering numbers in standard form.
Lakh is 5, crore is 7
Lakh has 5 zeros, crore has 7, arab has 9: odd numbers 5, 7, 9 and onward.
Use it when: Writing Indian large numbers as powers of 10.

Powers of 10 and Scientific Notation: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: A crore number in standard form · 1 mark · MCQBoard level

The standard form of $3, 57, 00, 000$ is: (a) $3.57 \times 1 0^{6}$ (b) $3.57 \times 1 0^{7}$ (c) $35.7 \times 1 0^{6}$ (d) $3.57 \times 1 0^{8}$

Try it first, then show the solutionSolution

$3, 57, 00, 000 = 35, 700, 000$ , which has 8 digits.
Put the point after the first digit and count: 7 places moved, so $3.57 \times 1 0^{7}$ .

Answer: (b) 3.57 x 10^7

Example 2: Expanded form with decimals · 2 marks · Short answerBoard level

Write $4072.65$ using powers of 10.

Try it first, then show the solutionSolution

Place values: 4 thousands, 0 hundreds, 7 tens, 2 units, 6 tenths, 5 hundredths. (1 mark)
$4072.65 = 4 \times 1 0^{3} + 0 \times 1 0^{2} + 7 \times 1 0^{1} + 2 \times 1 0^{0} + 6 \times 1 0^{- 1} + 5 \times 1 0^{- 2}$ . (1 mark)

Answer: 4 x 10^3 + 0 x 10^2 + 7 x 10^1 + 2 x 10^0 + 6 x 10^(-1) + 5 x 10^(-2)

Example 3: Small and large numbers · 2 marks · Short answerBoard level

Write in standard form: (i) $0.000062$ (ii) $8, 40, 000$ (iii) $0.5 \times 1 0^{- 3}$ .

Try it first, then show the solutionSolution

(i) Move the point 5 places right: $6.2 \times 1 0^{- 5}$ .
(ii) $840000$ : move 5 places left: $8.4 \times 1 0^{5}$ .
(iii) $0.5 \times 1 0^{- 3} = 5 \times 1 0^{- 1} \times 1 0^{- 3} = 5 \times 1 0^{- 4}$ .

Answer: (i) 6.2 x 10^(-5) (ii) 8.4 x 10^5 (iii) 5 x 10^(-4)

Example 4: Product in standard form · 3 marks · Short answerHarder board level

Find $(3 \times 1 0^{5}) \times (4.5 \times 1 0^{- 2})$ in standard form.

Try it first, then show the solutionSolution

Multiply coefficients: $3 \times 4.5 = 13.5$ . (1 mark)
Add exponents: $5 + (- 2) = 3$ , giving $13.5 \times 1 0^{3}$ . (1 mark)
Adjust the coefficient: $13.5 \times 1 0^{3} = 1.35 \times 1 0^{1} \times 1 0^{3} = 1.35 \times 1 0^{4}$ . (1 mark)

Answer: 1.35 x 10^4

Example 5: Time for a signal · 4 marks · Long answerHarder board level

Suppose a signal travels at $2.5 \times 1 0^{8}$ m/s over a distance of $1.5 \times 1 0^{11}$ m. Find the time in standard form and in minutes.

Try it first, then show the solutionSolution

Time $=$ distance $\div$ speed. (1 mark)
$\frac{1.5 \times 1 0 ^{11}}{2.5 \times 1 0 ^{8}} = 0.6 \times 1 0^{3}$ s. (1 mark)
Standard form: $0.6 \times 1 0^{3} = 6 \times 1 0^{- 1} \times 1 0^{3} = 6 \times 1 0^{2}$ s. (1 mark)
In minutes: $600 \div 60 = 10$ minutes. (1 mark)

Answer: 6 x 10^2 seconds, which is 10 minutes (hypothetical speed and distance).

Where students lose marks in Powers of 10 and Scientific Notation

The slips examiners see most often, and what to write instead.

Mistake: Leaving the coefficient outside 1 to 10
Standard form requires 1 <= x < 10.
Wrong working
" $13.5 \times 1 0^{3}$ is the answer."
Right working
$13.5 \times 1 0^{3} = 1.35 \times 1 0^{4}$ .
Do this: After multiplying coefficients, check the size of the result.
Usually costs: The final answer mark.
Mistake: Wrong sign of the exponent for small numbers
Numbers below 1 have negative exponents.
Wrong working
" $0.0035 = 3.5 \times 1 0^{3}$ "
Right working
$0.0035 = 3.5 \times 1 0^{- 3}$ .
Do this: Ask if the number is less than 1; if so the exponent is negative.
Usually costs: 1 mark.
Mistake: Miscounting the zeros in a crore or lakh number
Indian grouping (3,57,00,000) hides the digit count.
Wrong working
"3,57,00,000 = 3.57 x 10^6"
Right working
The number has 8 digits, so the exponent is $8 - 1 = 7$ : $3.57 \times 1 0^{7}$ .
Do this: Count digits, then subtract one for the exponent of a number above 1.
Usually costs: 1 mark.
Mistake: Adding exponents when dividing
Division subtracts the exponents.
Wrong working
" $\frac{6 \times 1 0 ^{8}}{2 \times 1 0 ^{3}} = 3 \times 1 0^{11}$ "
Right working
$\frac{6 \times 1 0 ^{8}}{2 \times 1 0 ^{3}} = 3 \times 1 0^{5}$ .
Do this: Multiply adds, divide subtracts.
Usually costs: The final answer.

Exam tips

Write the digits of the number, mark where the decimal point goes, and count the places before you write the exponent.
In estimation questions, state every assumption in a sentence and keep a unit with each number.
Do not round early; round the final coefficient to the requested significant figures.
Compare numbers by exponent first. Write the exponents next to each other in the comparison.
Use "about" or the approximation sign when your answer is an estimate.

Practice questions on Powers of 10 and Scientific Notation

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · Assertion-reasonBoard level
Assertion (A): 0.0035 = 3.5 x 10^(-3).
Reason (R): In scientific notation x times 10^y, the coefficient x satisfies 1 <= x < 10 and y is an integer.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Move the point three places right of 0.0035 to land after the 3.
Answer
(a). Moving the point 3 places to the right gives 3.5, so $0.0035 = 3.5 \times 1 0^{- 3}$ . R states exactly the rule that makes this the standard form.
Q2 · 3 marks · Short answerHarder board level
The Sun is $1.4335 \times 1 0^{12}$ m from Saturn and $1.496 \times 1 0^{11}$ m from Earth. About how many times as far is Saturn from the Sun as Earth is? Give your answer correct to one decimal place.
Hint
Divide and compare the powers of 10 separately.
Answer
$\frac{1.4335 \times 1 0 ^{12}}{1.496 \times 1 0 ^{11}} = \frac{1.4335}{1.496} \times 10 \approx 0.9582 \times 10 \approx 9.6$ . Saturn is about 9.6 times as far from the Sun as Earth is.
Q3 · 4 marks · Case-basedHOTS / competency
Case study: a sheet of paper is 0.001 cm thick. Each fold doubles the thickness. The Earth to Moon distance is 3,84,400 km.
(i) Write the thickness after n folds in cm using a power of 2. (1 mark)
(ii) Use 2^10 = 1024 to show that 2^45 is less than 3.52 x 10^13 and 2^46 is more than 6.4 x 10^13. (2 marks)
(iii) Show that 45 folds are not enough to reach the Moon but 46 folds are. (1 mark)
Hint
Write $2^{45} = 2^{5} \times (2^{10})^{4}$ , and use $102 4^{4} \approx 1.0995 \times 1 0^{12}$ . Remember 1 km $= 1 0^{5}$ cm.
Answer
(i) $0.001 \times 2^{n}$ cm. (ii) $(2^{10})^{4} = 102 4^{4}$ , which lies between $1.0 \times 1 0^{12}$ and $1.1 \times 1 0^{12}$ . So $2^{45} = 32 \times 102 4^{4} < 32 \times 1.1 \times 1 0^{12} = 3.52 \times 1 0^{13}$ and $2^{46} = 64 \times 102 4^{4} > 64 \times 1 0^{12} = 6.4 \times 1 0^{13}$ . (iii) For 45 folds the thickness is $< 0.001 \times 3.52 \times 1 0^{13} = 3.52 \times 1 0^{10}$ cm $= 3.52 \times 1 0^{5}$ km $< 3.844 \times 1 0^{5}$ km. For 46 folds it is $> 6.4 \times 1 0^{10}$ cm $= 6.4 \times 1 0^{5}$ km $> 3.844 \times 1 0^{5}$ km. So 46 folds are needed.
Q4 · 3 marks · Short answerHarder board level
The NCERT estimate of the number of trees on Earth is $3 \times 1 0^{12}$ and the human population is $8.2 \times 1 0^{9}$ . How many trees are there per person? Give the answer in standard form with two significant figures.
Hint
Divide the coefficients and subtract the exponents.
Answer
$\frac{3 \times 1 0 ^{12}}{8.2 \times 1 0 ^{9}} = \frac{3}{8.2} \times 1 0^{3} \approx 0.366 \times 1 0^{3} = 3.66 \times 1 0^{2} \approx 3.7 \times 1 0^{2}$ . About 370 trees per person.
Q5 · 4 marks · Long answerHOTS / competency
A year has 365 days. (i) Write the number of seconds in a year in standard form, correct to three significant figures. (ii) Show that $1 0^{9}$ seconds is about 31.7 years. (iii) How many seconds are there in 70 years, correct to two significant figures? (iv) Is $1 0^{6}$ seconds more or less than a fortnight? Explain.
Hint
365 x 24 x 3600 = 31,536,000.
Answer
(i) $365 \times 24 \times 3600 = 31, 536, 000 = 3.15 \times 1 0^{7}$ s. (ii) $\frac{1 0 ^{9}}{3.1536 \times 1 0 ^{7}} \approx 31.7$ years. (iii) $70 \times 3.1536 \times 1 0^{7} = 2.20752 \times 1 0^{9} \approx 2.2 \times 1 0^{9}$ s. (iv) $1 0^{6}$ s $\div 86400$ s per day $\approx 11.6$ days, which is less than a fortnight (14 days).

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What is the condition on the coefficient in scientific notation?Show answer
1 <= x < 10.
2.What power of 10 is a crore?Show answer
10^7.
3.What power of 10 is a lakh?Show answer
10^5.
4.What is the exponent of a number below 1 in standard form?Show answer
Negative.
5.What is a googol?Show answer
10^100.
6.Linear versus exponential growth?Show answer
Linear adds a fixed amount each step; exponential multiplies by a fixed factor each step.

Powers of 10 and Scientific Notation: frequently asked questions

Why do we write 5.9 x 10^3 and not 59 x 10^2?

Both equal 5900, but standard form fixes the coefficient between 1 and 10 so that numbers are easy to compare by their exponents.

What is the difference between a billion and an arab?

Both are 10^9: a billion in the international system and an arab in the Indian system.

What does the number of digits in the coefficient mean?

It shows how precisely the number is known: 1.42 x 10^5 claims more precision than 1.4 x 10^5.

Can the exponent be zero?

Yes. A number between 1 and 10 is written with exponent 0, for example 4.7 = 4.7 x 10^0.