Which of the following numbers is
represented by the arrow on the number line? Is it (A) the cube root of 12, (B)
the cube root of 15, (C) the cube root of 27, (D) the cube root of 45, or (E) the
cube root of 64?
In order to answer this question,
it is worth initially considering our cube numbers. In order to cube a number, we
multiply it by itself and itself again. This means that one cubed is equal
to one. Two cubed is equal to eight as two
multiplied by two is four, and multiplying this by two gives us eight. Three cubed is equal to 27. Continuing this list for the
integer values we have on our number line, we have 64, 125, 216, and 343.
Cube rooting is the opposite or
inverse of cubing. Therefore, the cube root of eight
is two. We can, therefore, match up the
radicals, the cube root of one, cube root of eight, cube root of 27, and so on, with
the integer values one to seven. The arrow on the number line lies
between three and four. This means that our answer must be
greater than the cube root of 27 and less than the cube root of 64. The only one of our five values
that lies between these two is the cube root of 45. The correct answer is option
(D). The cube root of 45 is greater than
three and less than four.
Options (C) and (E) cannot be
correct as they are equal to three and four, respectively. These are integer values;
therefore, the cube root of 27 and the cube root of 64 is rational. The cube root of 12 and the cube
root of 15 would both lie between two and three as 12 and 15 are greater than eight
but less than 27.