### Video Transcript

Hard drive A has a capacity of 3.979 times 10 to the power of 12 bytes. Hard drive B has a capacity which is 0.05 times the capacity of hard drive A. Part a) Calculate an estimate for the capacity of hard drive B, giving your answer in standard form. Part b) Is your answer to a) an overestimate or underestimate? Justify your answer.

So we’re gonna start part a. And the keyword is this word here, “estimate,” because what this means is that actually we’re gonna have to round our value before we start calculating, because if you find the exact value, you won’t get the marks. And this is a very common mistake. So what we’re gonna do is we’re gonna round 3.979 times 10 to the power of 12.

What we’re gonna round to is one significant figure, cause anytime we’re gonna do estimation, we use one significant figure. So therefore, we take a look at the first significant figure, which is three, and then we look at the number after it. And this number is actually a nine. So therefore, because it’s five or greater, it means we’re gonna round our three up to a four. So therefore, we can say that 3.979 times 10 to the power of 12 is equal to four multiplied by 10 to the power of 12 to one significant figure.

Okay, great! So now let’s use this for our calculation. So now to actually enable us to do the calculation, what we need to do is actually think about our 0.05 because what the question tells us is that hard drive B has a capacity which is 0.05 times the capacity of hard drive A.

Now we want to write this in standard form. Well, in order to actually write it in standard form, we write it in the form 𝑎 multiplied by 10 to the power of 𝑛, where our 𝑎 value must be greater than or equal to one or less than 10. So therefore, in this case, it’s gonna be five multiplied by 10 to the power of negative two.

But why is that? Well, the reason that that should be five multiplied by 10 to the power of negative two is because if we want to get from five to 0.05, then we’d have to actually divide by 100. And if we’re going to divide, then it means our power is actually going to be a negative power, because 10 to the power of negative two would actually mean five multiplied by one over 100. Well, it’d be five over 100, which would give us 0.05.

Okay, so now that we have four multiplied by 10 to the power of 12 and five multiplied by 10 to the power of negative two, what we can actually do is actually multiply these together to work out an estimate for the capacity of hard drive B. Now when we actually carry out this kind of calculation — so we’ve got four multiplied by 10 to the power of 12 multiplied by five multiplied by 10 to the power of negative two — what we actually do is actually multiply each part separately. So what this is gonna give us is four multiplied by five multiplied by 10 to the power of 12 multiplied by 10 to the power of negative two.

So therefore, this is gonna be equal to 20 multiplied by 10 to the power of 10. We get that because four multiplied by five is 20. And then 10 to the power of 12 multiplied by 10 to the power of negative two is 10 to the power of 10. And we get that because of one of our index laws. And that index law tells us that 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏.

So as we’ve got the same base, which is 10, then what we’re gonna do is add our powers. So if we have 12 add negative two, then we’re gonna subtract two from 12, which gives us 10. So we have 10 to the power of 10.

Okay, but have we finished? Well, if we take a look back at the question, we can see that it wants our answer in standard form. But is 20 multiplied by 10 to the power of 10 standard form? Well, the answer is actually no. And the reason it’s no is because if we look, our first part, so our 20, is not greater or equal to one or less than 10. So now we have to do one more Line to actually put it in standard form. So be careful at this point because actually a lot of students would stop at this point, and it’s the most common mistake where students will lose marks.

So therefore, this is gonna be equal to two multiplied by 10 to the power of 11. And that is gonna be a standard form. And we got that because what we did is we divided the 20 by 10 to give us two because that means that it satisfies the fact that I need it to be greater than or equal to one less than 10.

However, if we’ve divided that term by 10, then what we’re gonna have to do is actually multiply our 10 to the power of 10 by 10, which gives us 10 to the power of 11. So therefore, we can say that, given that hard drive A has capacity of 3.979 times 10 to the power of 12 and hard drive B has capacity which is 0.05 times the capacity of hard drive A, then our estimate for the capacity of hard drive B is two multiplied by 10 to the power of 11. And that’s in standard form.

Okay, now what we’re gonna do is actually move on to part b. Well, for part b, what we need to do is to actually just decide whether our answer to part a is actually an overestimate or an underestimate. Well, we can actually say that, in part a, we rounded 3.979 to four because we rounded to one significant figure. And 3.979 is actually less than four. So therefore, what we can say is that 3.979 was rounded up. So therefore, our answer to question a is an overestimate.