Video Transcript
Is the cube root of 27 a rational or an irrational number?
In this question, we are given an expression for a number and asked to determine if this number is rational or irrational. We can begin by recalling that we say that a number is rational if it can be written as the quotient of two integers. In other words, it must be equal to 𝑎 over 𝑏 for some integers 𝑎 and 𝑏, where 𝑏 is not equal to zero. An irrational number is a number that is nonrational, so any number that cannot be written in this form.
To determine if this number is rational or irrational, we can recall that the cube root of a number is the number whose cube gives that number. So if we say that the cube root of 27 is equal to 𝑥, then we must have that 𝑥 cubed equals 27. Then, by noting that 27 is equal to three cubed, we can see that 𝑥 must be equal to three. Since the cube root of 27 is three, the question then becomes, is three a rational or an irrational number?
We can show that three is rational either by writing it as three divided by one or by recalling that all integers are rational. Hence, the answer to this question is that the cube root of 27 is rational. We could have also answered this question by recalling that the cube root of a rational number is rational when the radicand is the quotient of perfect cubes, and it is irrational otherwise.
