PrepShorts · Study sheet · Class 8 Mathematics · Chapter 2, Power Play
Chapter 2 · Power Play
Zero and negative exponents: extending the rule rather than inventing a meaning
This video could not be loaded. Reload the page to try again.
Sign in with Google10 min.
Keep your place in this chapter — sign in, it’s free.Sign in
Nobody decided that 2⁰ should be 1. Keep erasing half of a line 16 units long and the value is forced on you.
The idea
Nobody decided that 2⁰ should be 1 because it looked tidy. The value is forced. Cancelling factors proves the division rule for counting exponents; that rule is then the only thing worth protecting, and protecting it leaves you no choice — 2⁰ has to be whatever 2⁴ ÷ 2⁴ is, which is 1, and 2⁻¹ has to be whatever 2⁴ ÷ 2⁵ is, which is a half. So a zero or negative exponent is not a new kind of multiplication that somebody had to invent a meaning for. It is the unique value that keeps one already-proved rule from breaking, and that is why the same symbols go on working past the edge of where they were defined.
What you should be able to do
- Derive the division rule for powers of a shared base by cancelling factors
- State the conditions the chapter attaches to that rule, and say why a zero base is excluded
- Explain why the value of 2⁰ is forced rather than chosen
- State that any nonzero base raised to the exponent 0 gives 1, and justify it
- Convert between a negative exponent and a reciprocal, in both directions
- Evaluate expressions mixing positive and negative exponents of one base
- Simplify a product or quotient of powers where the exponents are letters
- Say what question the chapter leaves open at the end of this section, and why it matters
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| generalised form | the chapter's name for a rule once it is written with letters | printed in this chapter (Part I pp.27–29) |
| counting numbers | the numbers 1, 2, 3, …, which is all the chapter's rules assume | printed in this chapter (Part I pp.27–28) |
| integers | the whole numbers together with their negatives, which is what the chapter asks about next | printed in this chapter (Part I p.29) |
| exponent | the count of factors, now allowed to be zero or negative | printed in bold in this chapter (Part I p.22) |
| base | the number being multiplied by itself, here barred from being 0 | printed in bold in this chapter (Part I p.22) |
| equivalent form | the chapter's phrasing for rewriting a negative exponent without one | printed in this chapter (Part I p.29) |
| reciprocal | 1 divided by a number | an added term; not printed in this chapter, which writes the fraction and never names it |
| extension by consistency | choosing a value so that an established rule keeps holding | an added term for the move this whole topic makes; not printed in this chapter |
Where people slip up
- "2⁰ is 0, because there is nothing to multiply." The empty product is 1, not 0, and the reason is the rule, not a convention about emptiness. Run 2⁴ ÷ 2⁴: nobody disputes that a number divided by itself is 1.
- "2⁻¹ is −2." The minus sits on the exponent, not on the value. A negative exponent of a positive base never produces a negative number. Put 2⁻¹ = ½ and −2 side by side.
- "A negative exponent means you divide by the base once." It means you divide as many times as the exponent says, starting from 1. 2⁻⁶ is one sixty-fourth, not one half.
- "0⁰ is 1 like everything else." The chapter's rules exclude a base of 0 every time, and the reason is exactly here: the derivation runs through division by xᵃ, which is illegal when x is 0. Answer the chapter's own Math Talk question rather than skipping it.
- "You can only divide a bigger power by a smaller one." That restriction is in the chapter's first statement of the rule, and the whole of this topic is about lifting it. Show why it was there and how it goes.
- "These are new definitions to memorise alongside the old ones." They are the only values consistent with the rule already proved. That is the argument, and an explanation that states 2⁰ = 1 without it has taught a fact instead of a reason.
- "1 over 10⁻³ needs a new rule." It needs the reciprocal of a reciprocal, which the chapter works out in two lines. Show the working.
Ask your teacher a person
Your teacher reads this and writes back, usually within a day. For an instant answer, use Ask the video in the sidebar.
Your class sees the question and the answer. Only your teacher sees that it was you.
No questions on this topic yet.
Worked answers to this chapter’s exercises · this video explains Figure it Out · 2.5 Q1, Figure it Out · 2.5 Q5
Transcript1,443 words
Here is a line, sixteen units long. Sixteen is two times two times two times two, so call it two to the fourth. Now erase half of it. Eight units left. Erase half of what is left. Four. Again: two. Nothing surprising has happened yet. But watch what the erasing is doing to the writing. Halving is dividing by two, and two is a power of two, so every step here is one power of two divided by another.
Take the third step on its own. Two to the fourth, divided by two cubed. You could try to remember a rule. Do not. Write both of them out. On the top, four 2s. On the bottom, three 2s. Now strike them off against each other, one for one, the way you would with any fraction. Three pairs go, and a single 2 is left standing on the top. The answer is two, and nothing was remembered.
Do the same for the other two steps and a pattern shows up. Four 2s over one 2 leaves three 2s: eight. Four 2s over two 2s leaves two 2s: four. Four 2s over three 2s leaves one 2: two. The count on top was four every time. The count on the bottom was one, then two, then three. And the count that survived was three, then two, then one.
It is a subtraction, and for a reason: striking off one pair removes one factor from the top and one from the bottom. So the exponents do not subtract because somebody said so. They subtract because that is what crossing things out does to a count. Written once, with letters: n to the a, divided by n to the b, is n to the a minus b. Try it on something you would not want to expand.
Two to the hundredth, divided by two to the twenty-fifth. A hundred minus twenty-five. Two to the seventy-fifth, and you never wrote a single 2 down. But look at the small print this rule arrives with, because the whole of the rest of this video is in it. The base is not allowed to be zero. Both counts have to be counting numbers: one, two, three, and up. And the count on top has to be the bigger one.
Every one of those conditions is there because of how the rule was got, not because of how it behaves. That third condition is doing something sneaky. It forbids a question. If the top count always has to be bigger, you can never write two to the fourth over two to the fourth and ask what the rule says. So ask it anyway. Four minus four is zero. What could two to the zero possibly be?
The honest answer is that at this moment it is not anything, because nobody has said what a row of no 2s means. You cannot multiply nothing together and read off an answer. There is no fact here waiting to be discovered. So we are not going to discover it. We are going to be forced into it. Here is the only lever available: the rule we already proved is worth keeping.
It came out of crossing factors off, and a rule that stops applying at one place is a bad rule. So whatever two to the zero turns out to be, it has to be whatever two to the fourth divided by two to the fourth is. And that one nobody argues about. Four 2s on the top, four 2s on the bottom, strike them all off, and the board is empty.
An empty board is one, not zero. A number divided by itself is one. So two to the zero is one, and notice what just happened: no definition was written, and no choice was made. The value was squeezed out. And there was nothing special about four. Two to the ninth over two to the ninth is one. Two to the six hundredth over two to the six hundredth is one.
Every single one of those differences is zero, and every single one of those answers is one. That is worth saying precisely. The rule does not merely allow two to the zero to be one. It leaves no other value on the table. If you tried to say it was zero, or two, or minus one, or a half, the rule would break somewhere. And there was nothing special about two either: any base you like, as long as it is not zero, raised to the exponent zero, is one.
Now go back to the sixteen-unit line and keep halving. Four halvings took it to one unit. Nothing stops you doing a fifth. Half a unit. By the rule, that is two to the fourth over two to the fifth, and four minus five is minus one. So there is the second forbidden question, and it arrived just by carrying on. What is two to the minus one? Same move as before, and the same lever.
Write it out: four 2s on the top, five 2s on the bottom. Strike off four pairs, and one lonely 2 is left on the bottom. One over two. A half. Two to the minus one is a half, and again nobody chose it. Keep going and the line keeps halving. Ten halvings from sixteen units. Four minus ten is minus six. Written out, four 2s over ten 2s, and after the striking there are six 2s left on the bottom.
One over sixty-four. So two to the minus six is one sixty-fourth. Not one sixth, and not one half. The exponent still counts factors; it has just moved to the other side of the line. Two things people get wrong here. The minus sits on the exponent, not on the answer. Two to the minus one is a half. It is not minus two. A positive base raised to any negative exponent is still positive, every time.
The second one: minus six does not mean divide once. It means divide six times. So the whole thing collapses into one short sentence. n to the minus a is one over n to the a. Ten to the minus three is one over a thousand. Seven to the minus two is one over forty-nine. And it reads backwards too, which surprises people: one over ten to the minus three is one divided by one thousandth, which is a thousand, which is ten cubed.
Put the three rules on one line and look at what you now have. Powers of the same base multiplied: the counts add. A power raised to a power: the counts multiply. Powers of the same base divided: the counts subtract. Two to the minus four times two to the seventh: minus four plus seven is three, and two cubed is eight. Three squared, times three to the minus five, times three to the sixth: two minus five plus six is three, and three cubed is twenty-seven.
Some equivalent forms: two to the minus four is one sixteenth, ten to the minus five is one over a hundred thousand, minus seven to the minus two is one over forty-nine, and minus five to the minus three is minus one over a hundred and twenty-five. The sign belongs to the base; the count belongs to the exponent. And one that looks like a trick until you write it out: two to the fourth times minus four to the minus two is sixteen times one sixteenth, which is one.
One condition is still sitting there unexplained: the base is not allowed to be zero. That is not fussiness. Look at where the value came from. Two to the zero is one because two to the fourth divided by two to the fourth is one. Run that with a base of zero. Zero to the fourth divided by zero to the fourth is nought over nought. There is no such number, so the argument does not merely give a strange answer; it never runs at all.
The forcing needed a division, and division by zero is not available, so nothing forces anything and the value stays genuinely undefined. That is the honest reason, and it is different from saying it is awkward. One thing is still open, and it is the right place to stop. Everything today started from a rule proved for counting numbers, and we have just spent the video using exponents that are neither counting numbers nor positive.
Whether all three rules really do survive at any whole number at all, plus or minus, is a question worth asking before you assume it.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- Exponential notation: repeated multiplication written onceClass 8 · Ch 2, Power Play
- The laws of exponents, and where each one comes fromClass 8 · Ch 2, Power Play
Comes up again in
- Reading a power line: multiplication as movement along a scaleClass 8 · Ch 2, Power Play
- Powers of 10, and place value written out for whole numbers and decimalsClass 8 · Ch 2, Power Play
- Scientific notation, and why the standard form is 1 ≤ x < 10Class 8 · Ch 2, Power Play
- Why the nearest power of ten is the only handle on a quantity too big to pictureClass 8 · Ch 2, Power Play