Powers of Ten

Study Sheet

Powers of Ten

What happens to the digits

Shifting Places

Concept
The Big Idea
1000100101110110011000373700
3.7 × 1000: the 3 and the 7 each move three places left; zeros fill the gaps.

Multiplying by 10n10^n moves every digit nn places to the left. The digits never change — only what each place is worth.

Multiplying by 1010 shifts every digit one place to the left; dividing by 1010 shifts it right. So 103=100010^3 = 1000 moves digits three places, and 3.7×103=37003.7\times10^3 = 3700. The exponent counts the shifts, which is why scientific notation can write huge and tiny numbers with a small exponent.

Example
Finding 43×10243 \times 10^2
1000100101110110011000434300
43 × 100: every digit two places left.

Two places left: 4343 becomes 43004300.

Example
What 10410^4 means

104=10×10×10×10=10,00010^4 = 10 \times 10 \times 10 \times 10 = 10{,}000: four factors of ten, so a 11 followed by four zeros. The exponent counts the tens being multiplied.

Side note
Where it goes wrong

Thinking "multiplying by 1010 adds a zero" in every case. That shortcut works for whole numbers but not for decimals: 3.7×103.7 \times 10 is not 3.703.70 (which is still 3.73.7); the digits shift to give 3737. Also, 100=110^0 = 1, not 00: zero shifts leaves a number unchanged.

Try it
Try it: powers and shifts

Write 10510^5 as a number, and find 7.2×1027.2 \times 10^2.

Show the answer

105=100,00010^5 = \mathbf{100{,}000} (five factors of ten). 7.2×1027.2 \times 10^2: every digit moves two places left, giving 720\mathbf{720}.

In Plain Terms & More Examples

Concept
In plain terms

Multiplying by 10n10^n slides every digit nn places to the left; the digits stay the same, only their value grows.

Example
Finding 6×1036 \times 10^3
100010010111011001100066000
The 6 moves from the ones to the thousands.

Three places left: 66 becomes 60006000.

Example
Finding 25×10225 \times 10^2

Two places left: 2525 becomes 25002500.

Side note
Where it goes wrong

Reading the exponent as a multiplier. 10310^3 is not 10×3=3010 \times 3 = 30; it is 10×10×10=100010 \times 10 \times 10 = 1000.

Try it
Try it: decimals and big shifts

Find 0.45×1030.45 \times 10^3 and 18×10418 \times 10^4.

Show the answer

0.45×1030.45 \times 10^3: three places left, 450\mathbf{450}. 18×104=180,00018 \times 10^4 = \mathbf{180{,}000}.

Dividing by Powers of Ten

Concept
Shift right

Dividing by 10n10^n moves every digit nn places to the right. Past the ones place you get decimals: 42÷103=0.04242 \div 10^3 = 0.042.

Example
Finding 6.5÷1026.5 \div 10^2
101110110011000650065
6.5 ÷ 100: both digits move two places right.

Two places right: 0.0650.065.

Example
Finding 3400÷1023400 \div 10^2

Two places right: 3400÷100=343400 \div 100 = 34. For a whole number ending in zeros, that is the same as removing two zeros.

Side note
Where it goes wrong

Shifting the wrong way when dividing. Dividing makes a number smaller, so the digits move right: 6.5÷102=0.0656.5 \div 10^2 = 0.065, not 650650.

Try it
Try it: both directions

Compute 0.37×1040.37\times10^4 and 815÷102815 \div 10^2.

Show the answer

0.37×104=37000.37\times10^4 = \mathbf{3700}. 815÷102=8.15815 \div 10^2 = \mathbf{8.15}.

Word Problems

Concept
Why scientists write powers of ten

Very large and very small numbers are easier to read as a number between 11 and 1010 times a power of ten: 384,000=3.84×105384{,}000 = 3.84 \times 10^5. The exponent tells you the size at a glance.

Counting the zeros in 384,000,000384{,}000{,}000 is slow and easy to get wrong; the 88 in 3.84×1083.84 \times 10^8 says it at once. When two numbers are written this way, the one with the bigger power of ten is the bigger number.

Example
Distance to the Moon

The Moon is about 3.84×1053.84\times10^5 km away — that is 384,000384{,}000 km. Writing 3.84×1053.84\times10^5 keeps the size clear without counting zeros.

Example
A stack of paper

A sheet of paper is about 0.10.1 mm thick. A stack of 10310^3 sheets is 0.1×1000=1000.1 \times 1000 = 100 mm tall, which is 1010 cm.

Side note
Where it goes wrong

Writing the front digits and then all the zeros. 3.84×1053.84 \times 10^5 is not 384384 followed by five zeros (38,400,00038{,}400{,}000). The point moves five places: 3.84→384,0003.84 \to 384{,}000.

Try it
Try it: cells

A drop of blood the size of a pinhead (one cubic millimetre) holds about 5×1065 \times 10^6 red blood cells. Write that as an ordinary number, and say how many cells are in 1010 such drops.

Show the answer

5×106=5,000,0005\times10^6 = \mathbf{5{,}000{,}000}. Ten drops: 5×107=50,000,0005\times10^7 = \mathbf{50{,}000{,}000}.

Patterns in the Zeros

Concept
Pull the tens aside

To multiply whole numbers that end in zeros, multiply the front digits, then attach all the zeros from both numbers: 40×300=4×3×10×100=12×1000=12,00040 \times 300 = 4 \times 3 \times 10 \times 100 = 12 \times 1000 = 12{,}000.

Each zero at the end of a factor is a factor of 1010, and multiplication can be done in any order. So the tens can be set aside, multiplied into one power of ten, and put back at the end. Powers of ten multiply by adding exponents: 102×10310^2 \times 10^3 has 2+3=52 + 3 = 5 tens.

Example
Finding 102×10310^2 \times 10^3
1010101010×1001000
Two tens and three tens: five tens in a row.

102×103=100×1000=100,000=10510^2 \times 10^3 = 100 \times 1000 = 100{,}000 = 10^5. Two tens times three tens is five tens multiplied together.

Example
Finding 40×30040 \times 300

4×3=124 \times 3 = 12. The factors end in 1+2=31 + 2 = 3 zeros, so 40×300=12,00040 \times 300 = 12{,}000.

Example
Finding 60×50060 \times 500

6×5=306 \times 5 = 30, which has a zero of its own. Attach the 33 zeros from the factors: 3030 followed by 000000 is 30,00030{,}000. The answer has four zeros.

Side note
Where it goes wrong

Counting only the zeros in the factors. 60×50060 \times 500 has three zeros in its factors, but 6×5=306 \times 5 = 30 adds a fourth: the answer is 30,00030{,}000, not 30003000. Multiply the front digits fully before attaching the zeros.

Try it
Try it: attach the zeros

Find 70×80070 \times 800 and 25×40025 \times 400.

Show the answer

7×8=567 \times 8 = 56, then attach 33 zeros: 56,000\mathbf{56{,}000}. 25×4=10025 \times 4 = 100, then attach 22 zeros: 10,000\mathbf{10{,}000}.

Why It Works

Each place in our number system is worth ten times the place to its right. Multiplying a number by 1010 therefore makes every digit worth ten times as much, which is the same as moving it one place left. Repeating this nn times is multiplying by 10n10^n. Division by 1010 is the reverse shift, and moving past the ones place produces decimals — the same rule, continued.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.