Write the number 8954000 in standard form.
Let’s recall what we mean by standard form. A number is in standard form if it looks like this: 𝑎 times 10 to the power of
𝑏. 𝑎 is a number that must be greater than or equal to one, but less than 10. 𝑏 must be an integer; that’s a whole number. And for very large numbers, 𝑏 will be a positive number. For small numbers — that’s numbers smaller than one — 𝑏 will be a negative
To write 8954000 in standard form, we first need to decide what the value of 𝑎 will
be. We’ll divide this number by 10 repeatedly until we have a number that satisfies the
criteria of being greater than or equal to one and less than 10.
If we divide this number by 10 six times — in other words if we divide by 10 to the
power of six — we end up with a number that’s between one and 10. It’s 8.954. This must mean then that in standard form 8954000 is the same as 8.954 times 10 to
the power of six.
Now, while it’s not hugely mathematical, there is a process that can help us remember
how to change a number into standard form. First, we decide where we want the decimal point to be to ensure that our number for
𝑎 is greater than or equal to one and less than 10. In this number, that’s between the eight and the nine.
Then, we find where the decimal point currently is, which for this number is at the
very end. We then count the number of jumps we need to make to get from where the decimal point
was to where it needs to be. In this example, that’s one, two, three, four, five, six jumps to get from where the
decimal point was to where we want it to be.
8954000 is a very big number. So when we write it in standard form, the power of 10 is positive. Once again, we can see that it’s 8.954 times 10 to the power of six.
Calculate the value of 7.8 times 10 to the power of seven divided by 2.4 times 10 to
the power of three. Give your answer in standard form.
When we perform multiplication and division with numbers that are in standard form,
we can rearrange our sum so that we’re dividing the leading numbers together. That’s what we called 𝑎 in the first part of this question. And we’re also dividing the powers of 10. In this case, that’s 7.8 divided by 2.4 all multiplied by 10 to the power of seven
divided by 10 to the power of three.
A more efficient way of writing this is to use the fraction symbols. 7.8 divided by 2.4 is 3.25. We’ll use the laws of indices to help us with the second part of this problem. Remember if we’re dividing two numbers with the same base, the base is usually the
big number. In our question, that’s the number 10. But in the general form, it’s the letter 𝑥.
We can subtract the powers. 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is 𝑥 to the power of 𝑎 minus
𝑏. That means then that 10 to the power of seven divided by 10 to the power of three is
equal to 10 to the power of seven minus three, which is four.
Our final answer in this question then is 3.25 multiplied by 10 to the power of
It’s useful to know that some calculators have a button that looks a bit like this
that will allow you to work out a question which is in standard form. It’s worth checking whether your calculator has one of these buttons. And if it does, practice using it. However, standard form questions like this could come up on a non-calculator
paper. So, it’s really useful to know the processes to take to answer it.