Video: GCSE Mathematics Foundation Tier Pack 3 • Paper 1 • Question 21

GCSE Mathematics Foundation Tier Pack 3 • Paper 1 • Question 21

07:38

Video Transcript

Calculate 0.005 times 0.0006 divided by 0.02. Give your answer in standard form.

Let’s start by looking at what standard form means. Standard form is a way in which we can easily write down very small or very large numbers. In order to do this, we write the number in the form of 𝑥 timesed by 10 to the power of 𝑛, where 𝑥 is a number such that 𝑥 is greater than or equal to one but less than 10 and 𝑛 is simply an integer, and 𝑛 can also be positive or negative.

Let’s quickly note that if 𝑛 is positive, then this means that we are multiplying 𝑥 by 10 𝑛 times. And if 𝑛 is negative, then we are dividing 𝑥 by 10 the positive of 𝑛 times. So if 𝑛 is equal to negative five, then we are dividing 𝑥 by 10 the positive of negative five times. That’s dividing 𝑥 by 10 five times.

Let’s look at another way in which we can consider these negative powers of 10. If we have 10 to the power of negative one, we can use an exponent rule which tells us that anything such as 𝑎 to the power of negative one is equal to the reciprocal of itself. So in this case, that’s one over 𝑎. Therefore, 10 to the power of negative one will be equal to one over 10. And this can also be written as 0.1. And so we can say that if we’re multiplying by 10 to the negative one, this is equivalent to multiplying by 0.1.

Similarly, if we consider 10 to the power of negative two, this is equal to one over 10 squared, and 10 squared is simply 100. So we can also write this as one over 100, which is also equal to 0.01. So multiplying by 10 to the negative two is equivalent to multiplying by 0.01. We can continue this for 10 to the negative three, which is equal to one over 10 cubed or one over 1000, which is also equal to 0.001.

Now we can use these pieces of information in order to help us write the numbers in our calculation in standard form. Starting with 0.005, we can see that 0.005 is equal to five times 0.001. And we’ve already shown that 0.001 is equal to 10 to the negative three. And so we can say that 0.005 is equal to five times 10 to the negative three. Since five is a number which is greater than or equal to one and less than 10 and negative three is an integer, five times 10 to the negative three is therefore in standard form. And so we’ve written 0.005 in standard form.

Let’s now move on to 0.0006. We can write this as six timesed by 0.0001. Now we haven’t actually found what 0.0001 is in terms of powers of 10. However, if we carry on our list one more time, we’ll see that 10 to the power of negative four is equal to one over 10 to the four or one over 10000, which is also equal to 0.0001. And 0.0001 is the same number as in our calculation. And since 0.0001 is equal to 10 to the negative four, we can say that 0.0006 is equal to six times 10 to the negative four.

Finally, let’s write 0.02 in standard form. We can say that 0.02 is equal to two times 0.01. And we already have that 0.01 is equal to 10 to the negative two. And so therefore 0.02 is equal to two times 10 to the negative two, which is in standard form.

So we found all three of our numbers in standard form. And so we can substitute these numbers in standard form into our calculation. This gives us five times 10 to the negative three times six times 10 to the negative four all divided by two times 10 to the negative two. And since it doesn’t matter which order we multiply in, we can move the numbers around in the numerator of the fraction to write it as five times six times 10 to the negative three times 10 to the negative four over two times 10 to the negative two. And now we’ve grouped up the powers of 10 in the numerator.

For the next step, we can spot that five times six is equal to 30. We can use an exponent rule which tells us that 𝑎 to the power of 𝑥 times 𝑎 to the power of 𝑦 is equal to 𝑎 to the power of 𝑥 plus 𝑦. And this will tell us that 10 to the power of negative three times 10 to the power of negative four is equal to 10 to the power of negative three plus negative four, which is also equal to 10 to the power of negative seven. Therefore, our fraction becomes 30 times 10 to the negative seven over two times 10 to the negative two.

Now in order to make this next step easier, we can split our fraction into two to get 30 over two times 10 to the negative seven over 10 to the negative two and evaluate each part individually. 30 over two is simply equal to 15. Then we need to use another exponent rule which tells us that 𝑎 to the power of 𝑥 divided by 𝑎 to the power of 𝑦 is equal to 𝑎 to the power of 𝑥 minus 𝑦. And this gives us that 10 to the power of negative seven divided by 10 to the power of negative two is equal to 10 to the power of negative seven minus negative two.

Now in this calculation, we’re subtracting a negative number. And when we subtract a negative number, this is equivalent to adding the positive of that number. So this becomes 10 to the power of negative seven plus two, which is also equal to 10 to the power of negative five. And so our fraction now becomes 15 times 10 to the negative five.

Now we’ve nearly finished our question. However, our answer is not in standard form. And the question requires our answer to be in standard form. Let’s recall our definition of standard form. We have 𝑥 times 10 to the power of 𝑛, where 𝑥 is greater than or equal to one and less than 10 and 𝑛 is an integer.

Now, as it stands in our calculation, our 𝑥 is 15, which is in fact greater than 10. And so our answer is not yet in standard form. In order to write 15 in standard form, we can write it as 1.5 times 10 to the power of one, since 1.5 is greater than or equal to one and less than 10 and one is an integer. So if we substitute this in here for 15, we get 1.5 times 10 to the one timesed by 10 to the negative five.

In order to simplify this, we need to use one of the exponent rules from earlier, which tells us that 𝑎 to the power of 𝑥 times 𝑎 to the power of 𝑦 is equal to 𝑎 to the power of 𝑥 plus 𝑦. This tells us that 10 to the power of one times 10 to the power of five is equal to 10 to the power of one plus negative five. And this is equal to 10 to the power of negative four.

And so we can say that our calculation is also equal to 1.5 times 10 to the negative four. We have that 1.5 is greater than or equal to one and less than 10, and negative four is an integer. Therefore, this answer is now in standard form. Therefore, our solution to the question is 1.5 times 10 to the negative four.

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