Video: AQA GCSE Mathematics Higher Tier Pack 1 • Paper 1 • Question 24

AQA GCSE Mathematics Higher Tier Pack 1 • Paper 1 • Question 24

03:59

Video Transcript

Write the square root of 32 plus 14 over the square root of two in the form 𝑎 root 𝑏, where 𝑎 and 𝑏 are prime numbers.

Whenever we see surds, those are square roots that don’t have a nice integer answer, we should always be thinking of the things we can do to manipulate them. There are several. But the two that spring to mind are simplifying a surd for the square root of 32 and rationalizing the denominator; that’s for the fraction.

Let’s begin then by simplifying the square root of 32. To do this, we need to find the largest factor of 32 that’s also a square number. Well, two of the factors of 32 are one and 32.

One is a square number. But actually, it’s not a particularly large one. 32 could be written as two multiplied by 16. 16 is a square number and it’s quite a large one. Let’s keep checking.

The final two factors for 32 are four and eight. And whilst four is a square number, it’s not as large as 16. So we choose the factor pair of 32 to be two and 16. And we can say that the square root of 32 is the same as the square root of two multiplied by 16.

At this point, we recall one of the rules that can be applied to surds. That is, the square root of 𝑎 multiplied by the square root of 𝑏 is the same as the square root of 𝑎 multiplied by 𝑏.

So as an example, the square root of two multiplied by the square root of five is the same as the square root of two multiplied by five, which is the square root of 10.

Now, what we’re going to do is reverse this rule and say that the square root of two multiplied by 16 can be written as the square root of two multiplied by the square root of 16.

But the square root of 16 is four. So we can say that the square root of two multiplied by the square root of 16 is the square root of two multiplied by four.

And a little like dealing with algebraic terms, we put the integer at the front. And we say that the square root of 32 can be written as four root two.

Now, let’s consider how we’re going to rationalize the denominator of 14 over root two. Currently, the denominator is an irrational number. That’s one that can’t be written as a fraction using integer values. So to rationalize it, we need to find a way to get rid of the square root.

And at this point, we remember that squaring is the opposite of square rooting. So if we can find a way to square the square root of two, that is, to multiply it by itself, the root will disappear.

In general, we can say that the square root of 𝑎 squared is equal to 𝑎 or we can say the square root of 𝑎 multiplied by the square root of 𝑎 is equal to 𝑎.

So we want to multiply the denominator by the square root of two. But we’re not allowed to do that just to the denominator. We have to do it to the numerator also. And the reason this works is because the square root of two over the square root of two is one.

And when we multiply by one, that doesn’t change the value of the fraction. Instead, we’re creating an equivalent fraction. And remember when we multiply two fractions, we multiply their numerators together and we multiply their denominators.

14 multiplied by root two is 14 root two. And we saw that the square root of two multiplied by itself is simply two. We can simplify this a little bit because both the numerator and the denominator share a factor of two. And that tells us that 14 over root two is equal to seven root two.

But how does this help us? Well, we’re going to rewrite our sum of root 32 plus 14 over root two as four root two plus seven root two. And if we have four root twos and we add seven more root twos, how many have we got? Well, we have 11 root twos. So four root two plus seven root two is equal to 11 root two.

And we’re done. We’ve written our expression in the form 𝑎 root 𝑏, where 𝑎 and 𝑏 are prime numbers. 𝑎 is 11 and 𝑏 is equal to two, both of which are prime numbers.

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