Video: Pack 2 • Paper 1 • Question 10

Pack 2 • Paper 1 • Question 10

06:08

Video Transcript

Write 0.000000902 in standard form.

Let’s first remind ourselves what standard form is. Standard form is a way in which we can easily write down very large or very small numbers. We do this by writing a number in terms of powers of 10.

Let’s, for example, write 4000 in standard form. We have that 4000 is equal to four times 1000. But then 1000 is also equal to 10 to the three. So we can write 4000 as four times 10 to the three. And this is now in standard form.

We can also write numbers smaller than one in standard form. Consider the number 0.00025. We could also write this as 2.5 times 0.0001. However, 0.0001 is the same as 10 to the power of minus four. And so we can say that 0.00025 is equal to 2.5 times 10 to the negative four. And this is now in standard form.

However, it’s important to note that the number which we’d multiply the power of 10 by — so in these two cases, it’s four and 2.5 — has to be greater than or equal to one and less than 10. This is because if we had a number out of this range, such as 0.9 or 11, we can write these as nine timesed by 10 to the negative one or 1.1 times 10 to the one. And these here are a correct standard form, since the numbers multiplying to the powers of 10 — so that’s nine and 1.1 — are between one and 10.

Before we write our number in standard form, let’s quickly note another way in which we can work out how to write a number in standard form. We can do this by counting how many places we have to move our decimal point in order to get our number in the range between one and 10.

So, for example, with 0.00025, we need to move the decimal point one, two, three, four places to the right. And so in our standard form, that’s how we know which power of 10 we need to use. So in this case, we moved it four places to the right, giving us the negative four.

Similarly, with 4000, our decimal point would appear after the last zero. So let’s count how many places we need to move this point in order to make our number between one and 10: one, two, three, places to the left. And so this is how we know that, in the standard form, we are raising the 10 to the power of three.

We can also note that if we’re moving the decimal point to the left, we get a positive power of 10. And if we’re moving the decimal point to the right, we get a negative power of 10.

Now we are ready to write our number in standard form. So let’s count how many places we move the decimal point in order to get a number between one and 10: one, two, three, four, five, six, seven places to the right. And doing this, the number which we end up with is 9.02. Then since we moved the decimal point seven places to the right, we need to multiply our 9.02 by 10 to the negative seven. And so this is 0.000000902 written in standard form.

Evaluate 2.4 times 10 to the seven over four timesed by 10 to the negative five. Give your answer in standard form.

First, we notice that we have a 2.4 in the numerator and a four in the denominator. And four does not divide 2.4. However, if we multiply 2.4 by 10, we get 24. And four does divide 24. So we can rewrite the numerator by taking one of the powers of 10 and multiplying it by the 2.4. And this gives us a new numerator of 24 times 10 to the six. So let’s substitute this into our fraction.

Now we have that our fraction is equal to 24 times 10 to the six over four times 10 to the power of negative five. Next, we will use the rule for multiplying fractions. This tells us that 𝑎 over 𝑏 timesed by 𝑐 over 𝑑 is equal to 𝑎𝑐 over 𝑏𝑑. We will use this in order to split our fraction up.

This tells us that our fraction is equal to 24 divided by four timesed by 10 to the six over 10 to the negative five. Then 24 divided by four is simply six. In order to simplify 10 to the six over 10 to the negative five, we can use an exponent rule.

We have that 𝑎 to the power of 𝑏 over 𝑎 to the power of 𝑐 is equal to 𝑎 to the power of 𝑏 minus 𝑐. And so we get that this is equal to six timesed by 10 to the power of six minus negative five. And then since subtracting a negative number is the same as adding the positive of that number, we end up with six timesed by 10 to the power of six plus five. And then six plus five is simply 11. And now we have six timesed by 10 to the 11. Since six is between one and 10, this number is now in standard form.

And so we have evaluated the fraction in the question, giving our answer in standard form, giving us a solution of six timesed by 10 to the 11.

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