There are four rules. If you multiply two powers of ten, you add the exponents. If you divide them, you subtract them. If you put a power over a power, you multiply. And a negative exponent is a fraction. The same four rules hold for units as well.
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Ten to the second times ten to the third. Two times three is six, so ten to the sixth? That is the mistake this material produces most often, and it comes back on every test where you calculate with large or small numbers.
In just over three minutes the four rules for powers of ten are set out: add when you multiply, subtract when you divide, multiply for a power over a power, and a negative exponent as a fraction. There are no more rules. After that, four sums step by step, with attention for the double minus sign where it goes wrong most. And the finest comes at the end: exactly these same four rules will soon hold for metres and for seconds, and then you calculate not with numbers but with units.
For upper secondary physics.
In this video
The full explanation, in writing
The problem
0:00Ten to the second, times ten to the third. Two times three is six, so ten to the sixth. Is that right, yes or no? Wrong. It is the mistake this material produces most often.
The four rules
0:16If you multiply two powers of ten, you add the exponents. Ten to the second times ten to the third is ten to the fifth. If you divide them, you subtract them. Ten to the p divided by ten to the q is ten to the p minus q. If you put a power over it, you multiply.
0:42Ten to the fourth, to the third, is ten to the twelfth. And a negative exponent is a fraction: ten to the minus p is one divided by ten to the p. There are no more rules. And the best is: exactly these same four rules will soon hold for metres, for seconds and for every other unit.
Four sums, step by step
1:07Four sums. Watch what happens to the exponents. Five divided by ten to the third is five times ten to the minus three. Twelve divided by four times ten to the third: first twelve divided by four is three, and then ten to the minus three. So three times ten to the minus three.
1:29Two point five times ten to the third, times three point zero times ten to the fourth: the numbers times each other is seven point five, the exponents added is seven. Seven point five times ten to the seventh. And six point three times ten to the minus five, divided by zero point ninety times ten to the minus two: the numbers divided is seven point zero, then the exponents: minus five minus minus two.
2:04Two minus signs next to each other become a plus, so minus five plus two is minus three. Seven point zero times ten to the minus three. Watch that double minus, that is where it goes wrong most. Four rules. Multiplying is adding, dividing is subtracting, a power over a power is multiplying, and a minus sign in the exponent is a fraction.
2:32And the finest is still to come: exactly these same four rules hold for metres and for seconds. Then you calculate not with numbers but with units. One question to take with you. Ten to the minus three, divided by ten to the minus five. Which exponent comes out of that?
2:53Minus three minus minus five. Two minus signs next to each other become a plus, so minus three plus five is two. Ten to the second. If you get minus eight, you have added the exponents instead of subtracting them. Four rules, and they will hold for units too.
3:11That is all there is to it.
Frequently asked questions
Do you add or multiply the exponents when you multiply two powers of ten?
You add them. Ten to the second times ten to the third is ten to the fifth, not ten to the sixth. Multiplying the exponents is the mistake this material produces most often.
How do you deal with two minus signs in an exponent?
Two minus signs next to each other become a plus. Minus five minus minus two is minus five plus two, so minus three. That double minus is where it goes wrong most.
How much is 6.3 times ten to the minus five divided by 0.90 times ten to the minus two?
Seven point zero times ten to the minus three. Divide the numbers first: 6.3 divided by 0.90 is 7.0. Then subtract the exponents: minus five minus minus two is minus three.
Do these rules hold for units too?
Yes, and that is the point of them. Exactly these same four rules hold for metres and for seconds. Metre times metre is metre squared, and metre per second may also be written as metre times second to the minus one.
What is a power over a power?
Then you multiply the exponents. Ten to the fourth, raised to the third, is ten to the twelfth. This is the only one of the four rules where multiplying is right.
Physics with Thomas makes physics explainers for upper secondary school. No words on screen, so what you see works in any language.
In this series
Quantities and unitswhy 20,000 is not more than 10,0007:30
Ratios and percentagesone table for both3:15
Scientific notationpowers of ten and order of magnitude5:22
Calculating with powers of tenthe four rules3:18
Pythagoras and trigonometrywhich side with which angle3:19
Deriving units from a formulaconverting to base units7:47
Significant figureshow many figures do you write6:24
Line of best fitthe line touches no point5:07
Relationships and the constantbraking twice as fast5:20
Linearising a graphyour measurements never change4:41
About me
My name is Thomas Schuurmans, and I have a passion for physics and for understanding why things work the way they do. What I want is to get young people asking “why is that?” and “how does that work?” more often, and to hand them the tools of physics in a way that is simple and that you can see.
I graduated in applied physics at Delft University of Technology in 2003. After that I spent more than twenty years outside education: first at TNO, the Dutch applied research institute, then at a design agency, and eventually founding Proportion Global, through which I work on innovation questions in Africa, Latin America and South Asia. That work resembles physics more than you would expect: don't start from a solution, first understand what is going on, try something, be wrong, and look again.
Since 2026 I have been teaching physics to both lower and upper secondary classes, and I started a master's at the University of Amsterdam for my full teaching qualification. That is where I learned that a secondary school pupil's real attention for new material lasts about seven minutes. And my own weakness happens to be telling too many side stories.
That is where the idea came from: videos that explain one topic sharply and visually, inside those seven minutes. I make them for my own pupils. Then I publish them, because good explanation should be within reach of anyone who needs it, wherever you live and whatever language you think and speak in.