Question Video: Identifying Irrational Numbers | Nagwa Question Video: Identifying Irrational Numbers | Nagwa

Question Video: Identifying Irrational Numbers Mathematics • Second Year of Preparatory School

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Which of the following is an irrational number? [A] ∛70 [B] ∛64 [C] 59 [D] √(144/81) [E] 109.5

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Video Transcript

Which of the following is an irrational number? Is it (A) the cube root of 70, (B) the cube root of 64, (C) 59, (D) the square root of 144 over 81, or (E) 109.5?

Let’s begin by recalling what we mean when we say a number is irrational. For a number to be rational, we must be able to write it in the form 𝑎 over 𝑏 where 𝑎 and 𝑏 are integers. They’re whole numbers. Then if a number is not able to be written in this form, it’s not rational. And in fact, we say it’s irrational.

So to work out which of our numbers is irrational, we’re going to go through them in turn. Let’s begin by looking at the cube root of 70. If we list out the cube numbers we know by heart, we see that we have three cubed is 27, four cubed is 64, and five cubed is 125. None of the numbers on the right-hand side are equal to 70. And this certainly tells us that the cube root of 70 doesn’t have a whole number in integer solution. It will be somewhere between four and five. It’s likely to be much closer to four since 70 is only a little bit bigger than 64.

So rather than trying to work out exactly what the cube root of 70 is equal to, we’ll look at the remaining four numbers. The cube root of 64, that’s (B), is actually in our list of cubic numbers. The cube root of 64 is equal to four. So actually, we can write it as four or four over one, meaning it is a rational number. And we can disregard (B) from our list. Let’s now look at (C). 59 is the same as 59 over one. Once again, both 59 and one are whole numbers. They’re integers. So we disregard (C). It’s also rational.

But what about (D)? Well, one of the rules we have for working with square roots is that we can find the square root of a fraction by finding individually the square root of the numerator and the square root of the denominator. The square root of 144 is 12, and the square root of 81 is nine. This means root 144 over 81 is rational. It is written in the form 𝑎 over 𝑏. 𝑎 is 12, and 𝑏 is nine. They’re both whole numbers. So what about (E)? Well, this is a terminating decimal. And in fact, we know that all terminating decimals are examples of rational numbers. It, therefore, cannot be irrational. And we’re going to disregard this one.

And so, by a combination of looking at the cube numbers and disregarding the other options, we see that the answer must be (A). The number that is irrational is the cube root of 70.

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