Number & Operations in Base Ten

Powers of Ten

Explaining patterns in the placement of a decimal point when multiplying by powers of 10.

Intermediate15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

What Really Happens When You Multiply by 10

Try predicting this before working it out: what is 4.7 multiplied by 10? By 100? By 1,000? Many people describe the answer as 'moving the decimal point,' but the decimal point doesn't actually move at all — something else is happening. What do you think is really going on?

Every digit's value depends entirely on its place — the same digit means ten times more one place to the left, and ten times less one place to the right. Multiplying by 10 doesn't move a decimal point; it shifts every digit one place value to the left, since each digit's value becomes ten times bigger. The decimal point only appears to move because the digits shifted underneath it.

Worked Example — Multiplying by a Power of Ten

Find 4.7 × 100. Multiplying by 10 shifts every digit one place to the left; multiplying by 100 (which is 10 × 10) shifts every digit two places to the left. The 4 moves from the ones place to the hundreds place, and the 7 moves from the tenths place to the tens place: 4.7 × 100 = 470.

Worked Example — Dividing by a Power of Ten

Find 358 ÷ 1,000. Dividing by 1,000 (which is 10 × 10 × 10) shifts every digit three places to the right, making each digit worth a thousand times less. 358 ÷ 1,000 = 0.358.

4.7 \times 10^{2} = 470 \qquad 358 \div 10^{3} = 0.358

Tip

Count the zeros in the power of ten to know how many places to shift — 10 has one zero (shift one place), 100 has two zeros (shift two places), and so on. Multiplying shifts left; dividing shifts right.

Common Mistakes

Thinking of the decimal point as physically moving through a fixed set of digits, and losing track of which direction it should go.
Think of the digits as shifting, not the decimal point — multiplying by a power of ten makes every digit represent a larger value, which shifts the whole number's digits to the left relative to the decimal point.
Miscounting the number of place-value shifts, especially for larger powers of ten like 1,000 or 10,000.
Count the zeros in the power of ten carefully — the count of zeros always equals the number of place-value shifts.

Key Takeaways

Multiplying by a power of ten shifts every digit that many places to the left; dividing shifts every digit that many places to the right.
This happens because each place value is exactly ten times the value of the place to its right.
The number of zeros in the power of ten tells you exactly how many places to shift.

Summary

Understanding powers of ten as a place-value shift, not a moving decimal point, builds real number sense. The next lesson extends decimal place value even further, to the thousandths place, and compares decimals there.

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