Powers of ten

Look at a lakh, a crore, or the distance to the Moon written out in full and the zeros start to blur together — did you count five zeros or six? Powers of ten fix this once and for all. Any power of ten, 10n10^n, is simply the digit 11 followed by nn zeros, and that one pattern turns a whole family of clumsy, error-prone numbers into something you can read and compare at a glance. This topic sits inside the Class 8 chapter "Power Play," and it is deliberately a bridge: it introduces no new law of exponents, but hands you the most familiar base of all — ten — on which to practise the multiplication, division and power-of-a-power laws you already learned, before those same laws travel to other bases and, soon after, into standard (scientific) form.

The core insight

10n10^n is the digit 11 followed by nn zeros, so multiplying by 10n10^n shifts every digit nn places up the place-value system, and the ordinary laws of exponents apply to base ten exactly as they do to any other base.

The core idea

A power of ten, 10n10^n, means 1010 multiplied by itself nn times:

10n=10×10×⋯×10⏟n times10^n = \underbrace{10\times10\times\cdots\times10}_{n\ \text{times}}

where: 1010 is the base and nn is the exponent (a whole number here). Each extra factor of 1010 shifts every digit one place to the left, which is exactly why the defining pattern holds: 10n10^n is the digit 11 followed by nn zeros.

101=10,102=100,103=1000,104=10,000,105=100,00010^1=10,\quad 10^2=100,\quad 10^3=1000,\quad 10^4=10{,}000,\quad 10^5=100{,}000

Because 10n10^n behaves like any other power, the three laws of exponents from earlier in the chapter carry straight over to base ten:

10m×10n=10m+n,10m÷10n=10m−n,(10m)n=10mn10^m\times10^n = 10^{m+n}, \qquad 10^m\div10^n = 10^{m-n}, \qquad (10^m)^n = 10^{mn}

where: mm and nn are whole-number exponents. These laws are what make it fast to combine powers of ten without ever writing out the zeros. Multiplying any whole number by 10n10^n has a shortcut that falls straight out of the pattern above: attach nn zeros to the end of the number (or, for a decimal, shift the decimal point nn places to the right). For example, 7×104=70,0007\times10^4 = 70{,}000 — shift 77 four places, i.e. attach four zeros. This is not a separate rule to memorise; it is just "10n10^n = 11 followed by nn zeros" combined with the fact that multiplying by 1010 once moves every digit one place value higher. Powers of ten also name large numbers compactly: a thousand is 10310^3, a lakh is 10510^5, a crore is 10710^7, a million is 10610^6, a billion is 10910^9. Notice that the Indian and international naming systems both rest on the very same base-ten pattern, they simply group the zeros differently in speech (lakh and crore versus million and billion) — the underlying power of ten is what makes the two systems intertranslatable at all, since converting between them is really just comparing exponents. This naming is the direct stepping stone to standard form, where any large or small number is written as k×10nk\times10^n with 1≤k<101\le k<10 — developed as its own topic right after this one, and every worked example below is really a small rehearsal for that skill. It also helps to see the pattern the other way round, going from a large ordinary number back to a power of ten. Given a round number such as 700,000700{,}000, first strip away the leading digit(s) — here 77 — and simply count the zeros that remain: five of them. That count is the exponent, so 700,000=7×105700{,}000 = 7\times10^5. This "count the trailing zeros" habit is the mirror image of the "attach nn zeros" shortcut, and mastering both directions together is what makes powers of ten fast and reliable rather than something to re-derive every time.

Figure
A number line style bar showing 10 to the power 1 up to 10 to the power 6 each labelled with its zero count growing by one step at a time
Each step of the exponent adds one zero 10^110 10^2100 10^31000 10^410,000 10^5100,000 Exponent up by one, one more zero, digits shift one place left 10 to the power n always equals a 1 followed by n zeros
Derivation AWhy the multiplication law works for powers of ten

The goal is to show, using nothing beyond the definition of a power as repeated multiplication and simple counting, why 10m×10n=10m+n10^m\times10^n=10^{m+n} genuinely holds for base ten rather than being a rule handed down without reason, so that the shortcut can be trusted rather than merely memorised.

Step 1. Write both powers out using the definition of 10m10^m and 10n10^n as repeated multiplication: 10m=10×10×⋯×10⏟m times10^m = \underbrace{10\times10\times\cdots\times10}_{m\ \text{times}} and 10n=10×10×⋯×10⏟n times10^n = \underbrace{10\times10\times\cdots\times10}_{n\ \text{times}}.

Step 2. Multiply the two expressions together: 10m×10n10^m\times10^n is now a string of 1010's multiplied together, with mm copies from the first power followed immediately by nn copies from the second power.

Step 3. Count the total number of 1010's in that combined string: there are mm copies plus nn copies, which is m+nm+n copies of 1010 multiplied together in total — and by the same definition, that is exactly 10m+n10^{m+n}.

Step 4. Conclude 10m×10n=10m+n10^m\times10^n = 10^{m+n}. The same counting argument, run in reverse for division (cancelling nn copies of 1010 from a string of mm copies leaves m−nm-n copies) and repeated for a power raised to a power (repeating a string of mm copies, nn times, gives mnmn copies), justifies all three laws for base ten.

Step 5. This counting argument is also exactly why the "zero-counting" pattern is trustworthy rather than a coincidence: since 10m10^m contributes mm zeros and 10n10^n contributes nn zeros, their product simply has m+nm+n zeros in total, matching 10m+n10^{m+n} digit for digit. The laws of exponents and the zero-counting shortcut are two views of the very same fact.

Worked example

Find the value of 4×1064\times10^6, express 2,000,0002{,}000{,}000 using a power of ten, and then simplify 108×103105\dfrac{10^8\times10^3}{10^5}.

Step 1. Recall 106=1,000,00010^6 = 1{,}000{,}000 (a 11 followed by 66 zeros), so 4×106=4×1,000,000=4,000,0004\times10^6 = 4\times1{,}000{,}000 = 4{,}000{,}000.

Step 2. Since 2,000,000=2×1,000,0002{,}000{,}000 = 2\times1{,}000{,}000, and 1,000,000=1061{,}000{,}000=10^6, we write 2,000,000=2×1062{,}000{,}000 = 2\times10^6.

Step 3. For 108×103105\dfrac{10^8\times10^3}{10^5}, apply the multiplication law to the numerator: 108×103=108+3=101110^8\times10^3 = 10^{8+3} = 10^{11}. Then apply the division law: 1011105=1011−5=106\dfrac{10^{11}}{10^5} = 10^{11-5} = 10^6.

Step 4. Write 10610^6 in ordinary form: a 11 followed by 66 zeros, so 106=1,000,00010^6 = 1{,}000{,}000.

Answer: 4×106=4,000,0004\times10^6 = 4{,}000{,}000; 2,000,000=2×1062{,}000{,}000 = 2\times10^6; and 108×103105=106=1,000,000\dfrac{10^8\times10^3}{10^5} = 10^6 = 1{,}000{,}000.

Competitive stretch

Express 50,000,00050{,}000{,}000 compactly as k×10nk\times10^n with 1≤k<101\le k<10. Since 50,000,000=5×10,000,000=5×10750{,}000{,}000 = 5\times10{,}000{,}000 = 5\times10^7, and 55 satisfies 1≤5<101\le5<10, the compact form is 5×1075\times10^7 — this is also exactly how standard form will write it later in the chapter.

Common traps

  • Miscounting the zeros when writing out 10n10^n. Writing 10510^5 as 10,00010{,}000 (four zeros) instead of the correct 100,000100{,}000 (five zeros) — always match the number of zeros to the exponent exactly.
  • Confusing n×10n\times10 with 10n10^n. For example, 104=10,00010^4=10{,}000, but 4×10=404\times10=40 — these look similar in words but are completely different calculations.
  • Forgetting to combine the leading number with the power of ten correctly, e.g. writing 3×1053\times10^5 as 3,00,0003{,}00{,}000 or misplacing digits — carefully attach the correct number of zeros after the leading digit(s).
  • Applying the laws of exponents to powers of ten mixed with other bases, e.g. wrongly simplifying 103×5310^3\times5^3 as 10610^6 or 50350^3 directly — remember 103×53=(10×5)3=50310^3\times5^3=(10\times5)^3=50^3 only because both exponents are the same (33); the bases combine, the exponents do not.
  • Losing track of place value when a power of ten is multiplied by a decimal, e.g. computing 2.5×1032.5\times10^3 by attaching three zeros to 2.52.5 literally instead of correctly shifting the decimal point three places to the right to get 25002500.
  • Trying to add or subtract exponents when the operation is addition of the numbers themselves, e.g. treating 103+10310^3+10^3 as 10610^6 or 103+310^{3+3} — the multiplication law only applies when the powers are being multiplied, not added; 103+103=2×103=200010^3+10^3=2\times10^3=2000, found by ordinary addition, not by combining exponents.

Ask the tutor: Mixed up how many zeros 10n10^n should have, or want a hand converting a big number into k×10nk\times10^n form? Ask the AI tutor to walk through it with you.

Why does multiplying by 10 to the power n just mean attaching n zeros?

Because 10n10^n itself is a 11 followed by nn zeros, and multiplying any whole number by that shifts every one of its digits nn places to the left in our place-value system — the same effect as attaching nn zeros to the end. If the number already has decimal digits, the same shift is easier to see as moving the decimal point nn places to the right rather than literally writing zeros after a decimal.

How is 10 to the power n different from n times 10?

They sound alike but are entirely different operations. n×10n\times10 is ordinary multiplication and just scales nn by ten, so 4×10=404\times10=40. 10n10^n is repeated multiplication of 1010 by itself nn times, so 104=10,00010^4=10{,}000. Always check which operation a question is actually asking for before writing an answer, since mixing the two is one of the most common slips with powers of ten.

Do the laws of exponents work differently for base ten than for other bases?

No — the laws 10m×10n=10m+n10^m\times10^n=10^{m+n}, 10m÷10n=10m−n10^m\div10^n=10^{m-n} and (10m)n=10mn(10^m)^n=10^{mn} are exactly the same laws you use for any base, just applied with base ten. Base ten only feels special because its powers double as the zero-counting shortcut for our place-value system; the underlying exponent rules never change from base to base.

Before this

Where this leads