Video: Estimating a Cube Root

Which of the following numbers is represented by the arrow on the number line? [A] ∛12 [B] ∛15 [C] ∛27 [D] ∛45 [E] ∛64

02:24

Video Transcript

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.

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