Calculating with numbers in standard form

Light travels three hundred million metres every second. A year is about thirty-two million seconds. Multiply those two and you get how far light travels in a year, in metres. Sixteen digits long. Nobody multiplies those out in longhand. Written in standard form they are three times ten to the power eight, and three point two times ten to the power seven. From there it is two small sums. Three times three point two is nine point six. Eight plus seven is fifteen. Nine point six times ten to the power fifteen metres. That is one light year, worked out in your head. That is what standard form buys you. But the powers only join up like that for multiply and divide. Try the same move on an addition and the answer comes out billions off. In this video you will handle all four operations, and see why plus and minus behave differently.

Multiplying first, because it is the friendliest of the four.

Work out four times ten to the power four, multiplied by three times ten to the power three. Give the answer in standard form.

Look at what you are actually multiplying: four numbers. Four, ten to the power four, three, and ten to the power three. Multiplication does not care what order you take them in. So gather the front numbers together, and the powers of ten together.

Front numbers first. Four times three is twelve.

Now the powers. Ten to the power four times ten to the power three is ten to the power seven. Four tens multiplied, then three more tens multiplied, is seven tens multiplied. That is the rule from the video on index laws: multiplying and dividing powers.

So the answer so far is twelve times ten to the power seven. The value is right. The form is not.

This is the same range check as the video on converting to and from standard form. One digit before the point. Twelve has two, so this is not finished.

Twelve is one point two times ten. So twelve times ten to the power seven is one point two, times ten, times ten to the power seven. That is one extra ten to carry into the power. The answer is one point two times ten to the power eight.

Here is the line to keep hold of. Fronts with fronts, powers with powers.

Your turn. Work out two times ten to the power three, multiplied by four times ten to the power six. Is it A, six times ten to the power nine. B, eight times ten to the power nine. Or C, eight times ten to the power eighteen.

Have a think. I'll wait.

The answer is B. Eight times ten to the power nine.

Two times four is eight, so the front is eight, not six. A added the fronts instead of multiplying them. Three plus six is nine, so the power is nine, not eighteen. C multiplied the powers instead of adding them. Each pair gets its own sum.

Now dividing. Same split of the work, opposite sums.

Work out nine times ten to the power eight, divided by three times ten to the power five.

Fronts with fronts: nine divided by three is three. Powers with powers: ten to the power eight divided by ten to the power five.

Dividing by ten to the power five cancels five of those eight tens. Three tens are left standing. That is why the powers subtract. Eight take away five is three.

The answer is three times ten to the power three. Three thousand.

Run the range check anyway. Three is one digit before the point, so there is nothing to tidy. The multiplying answer needed fixing and this one does not. Same check, different outcome, which is why you run it every time.

Your turn. Work out six point three times ten to the power six, divided by three times ten to the power three.

Pause it there and work it out. I'll wait.

The answer is two point one times ten to the power three.

Six point three divided by three is two point one. Six take away three is three. And two point one already has one digit before the point, so it is finished as it stands.

Now plus and minus. These two do not behave like the other two.

Work out five times ten to the power five, plus three times ten to the power four. The tempting move is to copy the multiply rule. Five plus three is eight. Five plus four is nine. Eight times ten to the power nine.

Eight times ten to the power nine is eight billion. The true answer is just over half a million. That is not a small slip. It is out by a factor of about fifteen thousand.

Write both numbers out instead. Five times ten to the power five is five hundred thousand. Three times ten to the power four is thirty thousand. Add them and you get five hundred and thirty thousand. In standard form, five point three times ten to the power five.

Here is why the powers cannot simply join. Multiplying is about how many tens you have got, so the tens pile up. Adding is about place value. Five hundred-thousands and three ten-thousands are different-sized units. You would not add five metres to three centimetres and call the answer eight of anything.

For numbers too big to write out, match the powers first. Five times ten to the power five is fifty times ten to the power four. Now both numbers are lots of ten to the power four, the same size of unit. Fifty plus three is fifty-three, so fifty-three times ten to the power four. The range check then tidies that to five point three times ten to the power five. Same answer.

When the powers already match, it is quicker still. Four point three times ten to the power minus four, plus eight point one times ten to the power minus four. Both are lots of the same unit, so just add the fronts. Four point three plus eight point one is twelve point four.

Twelve point four times ten to the power minus four. Twelve point four has two digits before the point, so it tidies to one point two four times ten to the power minus three. The check earns its keep on small numbers too.

So the second line to keep hold of. Plus and minus, powers don't play.

Your turn. Work out seven times ten to the power five, take away two times ten to the power four. Answer in standard form.

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

The answer is six point eight times ten to the power five.

Seven hundred thousand take away twenty thousand is six hundred and eighty thousand. Or match the powers: seventy times ten to the power four, take away two times ten to the power four, is sixty-eight times ten to the power four. Either route tidies to six point eight times ten to the power five.

Last stop. What the exam paper is actually asking for when it says standard form.

Giving your answer in standard form is an instruction about your final line, not your working. Twelve times ten to the power seven is the right value in the wrong form. Tidying it is part of the question. On a calculator paper the machine does the arithmetic, in its own shorthand: one point two, then a capital E, then eight. That E stands for times ten to the power. Write it on the page as one point two times ten to the power eight. The E is calculator shorthand, not standard form. On a non-calculator paper, put the two sums on separate lines. Fronts on one line, powers on the other. Keeping them apart is what stops you doing the powers' sum to the front numbers by accident.

Time to bring the fronts and the powers back together.

Multiplying: multiply the fronts, add the powers. Dividing: divide the fronts, subtract the powers. Two parts of the number, two separate sums. Adding and subtracting: the powers do not combine at all. Write the numbers out in full, or match the powers first, then add or subtract the fronts. And every answer gets the same check. One digit before the point. If the front number has grown to ten or more, or shrunk below one, the answer is not finished.

Two lines to carry into the exam hall. Fronts with fronts, powers with powers. Plus and minus, powers don't play.

Next in the chapter: solving problems in standard form, which is Higher tier only. If you are sitting Foundation, standard form finishes here, with this video.

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