Video: AQA GCSE Mathematics Higher Tier Pack 3 • Paper 1 • Question 22

The expression 18/√3 − 45/√75 + 4√27 can be written in the form 𝑎√3, where 𝑎 is an integer. Find the value of 𝑎. Show all your working.

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Video Transcript

The expression 18 divided by root three minus 45 divided by root 75 plus four root 27 can be written in the form 𝑎 root three, where 𝑎 is an integer. Find the value of 𝑎. Show all your working.

In order to answer this question, we need to simplify each of the three terms individually. When simplifying each of the terms, we will need to use two of the laws of surds: root 𝑎 multiplied by root 𝑎 is equal to 𝑎 and root 𝑎 multiplied by root 𝑏 is equal to the root of 𝑎 multiplied by 𝑏.

Let’s consider the first term 18 divided by root three. We need to rewrite this in the form 𝑎 root three. In order to simplify this term, we need to rationalize the denominator. Rationalizing the denominator involves getting rid of the surds from the denominator.

Remember with fractions whatever we do to the top, we must do to the bottom. In this case, we’ll multiply the numerator and denominator by root three. 18 multiplied by root three is equal to 18 root three and root three multiplied by root three is equal to three, as shown in our first law of surds: root 𝑎 multiplied by root 𝑎 is equal to 𝑎. As 18 divided by three is equal to six, the first term 18 divided by root three is equal to six root three.

We now need to consider the second term 45 divided by root 75. One way to simplify this would be to multiply the numerator and denominator by root 75. An alternative method to make our arithmetic simpler would be to simplify root 75 first. Using the second law of surds, we can rewrite root 75 as root 25 multiplied by root three as 25 multiplied by three is equal to 75.

The reason we chose 25 as one of the factors is that 25 is a square number. This means that we can square root it and get an integer value or whole number answer. The square root of 25 is equal to five. Therefore, root 25 multiplied by root three can be rewritten as five multiplied by root three or five root three.

We can, therefore, rewrite the second term as 45 divided by five root three as root 75 is equal to five root three. 45 divided by five is equal to nine. So this can be simplified to nine divided by root three.

At this point, we need to rationalise the denominator. We need to multiply the top and bottom by root three. Nine multiplied by root three is equal to nine root three. And root three multiplied by root three is equal to three. Finally, nine divided by three equals three. So the second term 45 divided by root 75 can be simplified to three root three.

We now need to consider the third term four root 27 and try and write this in the form 𝑎 root three. Our first step here is to simplify root 27. 27 is equal to nine times three. Therefore, root 27 is equal to root nine multiplied by root three. The square root of nine is three. Therefore, root 27 is equal to three root three.

This means that four root 27 can be written as four multiplied by three root three. As four times three equals 12, four root 27 is equal to 12 root three. We have now simplified all three terms so they are in the form 𝑎 root three. 18 divided by root three minus 45 divided by root 75 plus four root 27 is equal to six root three minus three root three plus 12 root three.

Six minus three is equal to three and three add 12 equals 15. Therefore, six root three minus three root three plus 12 root three is equal to 15 root three. Our expression is now written in the form 𝑎 root three, where 𝑎 is equal to 15.

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