Mathematics from zero
Powers of ten
Ten, a hundred, a thousand, a million. Each is ten times the one before. They are not random big numbers — they are one number, 10, raised to higher and higher powers. And that pattern is the quiet engine behind everything you learned about place value.
After this lesson you can compute any power of ten, explain why its exponent equals its number of zeros, connect powers of ten to place value, and write a large round number as a small number times a power of ten.
A power of ten is 10 multiplied by itself. 10² is 10 × 10 = 100. 10³ is
10 × 10 × 10 = 1000. The base is always 10; only the exponent changes. These are the
powers of ten, and they are the most regular numbers in all of arithmetic.
The exponent is exactly the number of zeros. 10² is 100 — two zeros. 10³ is
1000 — three zeros. 10⁶ is 1000000 — six zeros. Each time you raise the power by
one, you multiply by 10, which writes one more zero on the end. So to find 10ⁿ, write
a 1 and then n zeros.
Powers of ten are what place value is built from. Reading a number from the right,
each place is a power of ten. The ones place is 10⁰ = 1, the tens place is
10¹ = 10, the hundreds place is 10² = 100. The “each place is ten times the last”
rule you learned for place value is just the powers of ten in a row.
A large round number can be written as a small number times a power of ten. 3000
is 3 × 1000, which is 3 × 10³. 60000 is 6 × 10⁴. This scientific form —
a small number times a power of ten — keeps very large numbers short and easy to
compare: you read the power of ten first to see the size, then the small number for the
detail.
Write 50000 as a small number times a power of ten.
First count the zeros in 50000: there are four. So this number is 5 followed by
four zeros.
Four zeros means the power of ten is 10⁴ (which is 10000).
So 50000 = 5 × 10⁴. Check it: 10⁴ = 10000, and 5 × 10000 = 50000. The small
number 5 carries the leading digit; the 10⁴ carries the size.
Why this works
Why does raising the power by one add exactly one zero? Because raising the power by
one multiplies by 10 once more — and multiplying any whole number by 10 shifts every
digit one place to the left, dropping a fresh 0 into the ones place. 100 × 10 = 1000.
The zero count rises in lockstep with the exponent because each is counting the same
thing: how many times you multiplied by ten.
Common mistake
A common mistake is thinking 10⁴ has four as a digit somewhere, or equals 40. It
does not: 10⁴ is 10000. The exponent tells you the number of zeros, not a digit
in the answer and not a factor to multiply 10 by. 10⁴ means four tens multiplied
together.
Work out 10². Type the value.
Work out 10⁴. Type the value.
How many zeros does 10⁶ have? Type the count.
Work out 10⁰. Type the value.
Work out 5 × 10². Type the value.
Without multiplying it out, how many zeros does 10⁵ have?
A power of ten is 10 multiplied by itself, and its exponent is exactly its number of
zeros — 10ⁿ is a 1 followed by n zeros. Powers of ten are the engine of place value:
each place, from the right, is the next power of ten. And any large round number can
be written compactly as a small number times a power of ten, which keeps big numbers
short and easy to size up.