Index laws: multiplying and dividing powers

A message goes round your year group. Everyone who gets it sends it on to three new people. Round one, three people. Round two, nine. By round six, seven hundred and twenty-nine people get it in that round alone. That is three to the power six. It keeps running for five more rounds. Five more rounds means multiplying by three another five times. That is three to the power five, which is two hundred and forty-three. So round eleven is three to the power six, times three to the power five. There is a one-line route to that. Add the little numbers. Six plus five is eleven. Three to the power eleven. A hundred and seventy-seven thousand, one hundred and forty-seven. The tempting wrong move is to multiply the threes as well and write nine to the power eleven. That is over thirty-one billion. Nearly four times as many people as there are on the planet.

So, why those little numbers add. Get the reason and the rule stops being something to remember.

Take five to the power four, multiplied by five to the power two. Write both of them out in full. Five to the power four is four fives multiplied together. Five to the power two is two more fives. Push them into one line and you have six fives multiplied together. That is five to the power six.

Now look at what is actually in that line. Every factor in it is a five. Nothing anywhere in there made a twenty-five. So the base cannot change. The only thing that changed is how many copies there are. Four and two became six.

That is the first index law. Same base, multiplied together: add the indices. It only works when the two bases match. Watch it on numbers you already know. Eighty-one is four threes, so three to the power four. Nine is three to the power two. Multiply them and you have six threes. Three to the power six. Seven hundred and twenty-nine, the message chain again.

Here is the phrase to carry into the exam hall. Count the copies. The base holds still. In the video on square roots and cube roots, the phrase was that the little number counts the copies. This is that idea doing a job.

Your turn, and it is one step. Which of these is two to the power three, multiplied by two to the power four? Two to the power seven. Four to the power seven. Or two to the power twelve.

Have a think. I'll wait.

The answer is two to the power seven. Three copies of two, then four more copies, makes seven copies. Four to the power seven has multiplied the bases as well, and the base holds still. Two to the power twelve has multiplied the little numbers instead of adding them.

Check that against numbers you can do by hand. Two to the power three is eight. Two to the power four is sixteen. Eight times sixteen is a hundred and twenty-eight. And two to the power seven is a hundred and twenty-eight. Same answer, one step instead of three.

Now dividing. Same idea, running the other way.

Work out six to the power nine, divided by six to the power seven. No calculator.

The instinct is to find both numbers first. Six to the power nine is over ten million. Six to the power seven is two hundred and seventy-nine thousand, nine hundred and thirty-six. That is two long multiplications before the division has even started. It is where most people stall.

Count copies instead. Nine sixes on the top. Seven sixes on the bottom. Every six underneath cancels a six above it. Seven pairs go, and two sixes are left standing on the top. Six to the power two. Thirty-six.

So the second law. Same base, divided: subtract the indices. Nine take away seven is two. Tens make it obvious. Ten to the power five, divided by ten to the power two, is ten to the power three. A thousand.

And the base survives the division too. Six divided by six is one, so it is tempting to write one to the power two. But nothing in that calculation is a one. Seven pairs cancelled each other out, and two whole sixes were left. The answer is thirty-six.

Examiners watch the long route fail. Their words, from a Foundation paper report:

This was a standard style of working with indices but the majority of students started by trying to find the actual value of eight to the power three and eight to the power four. They were then unable to process the necessary division and hadn't written down that they planned to attempt a division, so were unable to score beyond the first mark.

Finding the two values was not the hard part. Getting through the division afterwards was, and with no division written down there was nothing left to earn a mark on. Counting copies never builds those numbers at all.

Now both laws in one calculation, which is how questions usually turn up.

Simplify five to the power four, multiplied by five to the power three, all divided by five to the power two. Leave your answer as a power of five. Work left to right, and keep the base still.

Have a go at this one. I'll wait.

The answer is five to the power five. Four plus three is seven, so the top is five to the power seven. Then seven take away two is five. In one line: four, plus three, take away two, is five.

In copies: seven fives multiplied on the top, two of them cancelled from underneath, five fives left standing. One base, one running count, all the way through.

Now the exam side. There is one mistake examiners name directly. Their words, from a Higher paper report:

On question 11, many students multiplied the bases as well as adding the indices when multiplying, eg saying that three to the power six times three to the power five equals nine to the power eleven, similarly when dividing.

The adding was right. The bases came along for the ride. And look at what that costs. Three to the power eleven is about a hundred and seventy-seven thousand. Nine to the power eleven is thirty-one billion. The indices were correct, and the answer is out by a factor of a hundred and seventy-seven thousand.

So here is a habit for before you write anything. Check the bases. Only one of these two can be written as a single power. Four to the power three, times four to the power two. Or four to the power three, times three to the power two.

Which one? I'll wait.

The first one. Both bases are four, so count the copies. Three plus two is five. Four to the power five. The second one has a four and a three. Different bases, nothing shared to count, so it will not fold into a single power at all.

One last piece of wording. Leave your answer as a power of five. That is an instruction. Five to the power six is what is being asked for. Grinding it out to fifteen thousand, six hundred and twenty-five answers a different question.

So, both laws in one place now.

Two laws, one idea. Powers of the same base multiplied together: add the indices. Five to the power four times five to the power two is five to the power six. Divided: subtract them. Six to the power nine divided by six to the power seven is six to the power two, which is thirty-six. Both of them are counting copies, and in every line the base holds still. A six on the top, a six on the bottom, a six in the answer. If the base has changed, something has gone wrong. And if the two bases were different to start with, there is nothing to combine.

Next in this chapter: power of a power, and zero and negative indices. What happens when a power is raised to another power, and what an index of zero or below actually means.

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