Multiplying and Dividing by Powers of 10

Why this matters

Our number system is built on tens. Every place is worth ten times the place to its right, which is why multiplying by 10 simply slides the digits along. Understanding this is the foundation for standard form, metric conversions and every calculation with very big or very small numbers.

The pattern

Each time you step down this list, you divide by 10:

Power	$10^3$	$10^2$	$10^1$	$10^0$	$10^{-1}$	$10^{-2}$	$10^{-3}$
Value	1000	100	10	1	$\frac{1}{10} = 0.1$	$\frac{1}{100} = 0.01$	$\frac{1}{1000} = 0.001$

That pattern tells you everything:

multiplying by $10^{-1}$ is the same as dividing by 10,
multiplying by $10^{-2}$ is the same as dividing by 100,
dividing by $10^{-3}$ is the same as multiplying by 1000,
and in general, dividing by $10^{-6}$ is multiplying by $10^6$ .

Digits move, the decimal point stays

It helps to picture the digits sliding past a fixed decimal point:

$2.87 \times 10^3 = 2870 \qquad 2870 \times 10^{-3} = 2.87$

Fill any empty places with zeros.

Sense check. Multiplying by a positive power of 10 makes a number bigger. Multiplying by a negative power of 10 makes it smaller. If your answer goes the wrong way, you have flipped the operation.

Order of operations still applies

In $7 \times 10^5 + 6 \times 10^3$ , work out each product first, then add: $700\,000 + 6000 = 706\,000$ .

Key rules and formulas

Positive powers

$3 \times 1 0^{4} = 3 \times 10000 = 30000$

Multiplying by

1 0^{n}

moves every digit

n

places to the left (the number gets bigger). Dividing by

1 0^{n}

moves them

n

places to the right.

Negative powers flip the operation

$\times 1 0^{- n} = \div 1 0^{n} \div 1 0^{- n} = \times 1 0^{n}$

Because

1 0^{- 2} = \frac{1}{100}

, multiplying by

1 0^{- 2}

divides by 100, and dividing by

1 0^{- 2}

multiplies by 100.

Link to 0.1 and 0.01

1 0^{- 1} = 0.1

1 0^{- 2} = 0.01

1 0^{- 3} = 0.001

. So multiplying by 0.01 is the same as dividing by 100.

Worked examples

Worked example 1

Four calculations

Work out a

3 \times 1 0^{4}

420 \div 1 0^{2}

530 \times 1 0^{- 3}

0.819 \div 1 0^{- 5}

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Finding a missing power

Find the missing power:

60000 \times 1 0^{□} = 6

Try it yourself first, then reveal the solution one step at a time.

Calculation	Missing number
7.1 × 10^□ = 710 000	Row 1, Missing number
□ ÷ 10³ = 249	Row 2, Missing number
0.02 ÷ 10^□ = 2	Row 3, Missing number
44 000 ÷ 10^□ = 4.4	Row 4, Missing number
□ × 10⁻² = 2.3	Row 5, Missing number
□ ÷ 10⁻⁶ = 7000	Row 6, Missing number

Calculation

Missing number

7.1 × 10^□ = 710 000

□ ÷ 10³ = 249

0.02 ÷ 10^□ = 2

44 000 ÷ 10^□ = 4.4

□ × 10⁻² = 2.3

□ ÷ 10⁻⁶ = 7000