Powers of Ten

10 × 10 × 10 is written 10³. The small 3 counts how many tens are multiplied.

The small raised number has a name.

Exponent says how many times a number is multiplied by itself.

Counting the zeros

10³ is 1,000, which has 3 zeros. The exponent and the zeros always match.

A warning

10³ is not 30. It is 10 multiplied three times, not 10 added three times.

What the small raised number means

10³ means 10 × 10 × 10 = 1,000. The raised number, called the exponent, counts how many tens are multiplied together — not what to multiply by. So 10⁵ is 100,000, a 1 followed by five zeros. The exponent and the zero count match for powers of ten, which is a useful coincidence but only a coincidence.

Why a shorthand is worth having

Writing 1,000,000,000 is error-prone; writing 10⁹ is not. Scientists dealing with distances between stars or the size of atoms would be lost without it. The notation also makes patterns visible: multiplying 10³ by 10² gives 10⁵, because you are simply multiplying five tens together.

Expanded form with powers

3,406 can be written (3 × 10³) + (4 × 10²) + (0 × 10¹) + (6 × 10⁰). This says exactly what place value has always said, in a notation that scales to any size of number. And 10⁰ = 1, which follows from the pattern: each step down divides by ten, and 10¹ = 10 divided by ten is 1.

Multiplying and dividing by powers of ten

Multiplying by 10² shifts every digit two places left; dividing by 10² shifts them two places right. Thinking of it as digits moving rather than zeros being added is what keeps the rule working once decimals are involved, where "add a zero" fails outright.