Which of the following numbers is a cubic root of a whole cube between 1 and 1000? A) 11 B) 10 C) 8 D) 9

Understand the Problem

The question is asking which of the provided numbers is a cubic root of a whole number that is the cube of an integer within the range of 1 to 1000. We will look for whole cubes in that range and determine their cubic roots.

Answer

The cubic roots of whole numbers from 1 to 1000 are: $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$.

Steps to Solve

Identify the range of numbers The problem specifies that we are looking for whole cubes of integers within the range of 1 to 1000. We need to find integers $n$ where $n^3 \leq 1000$.
Calculate integer cubes within the range We will calculate the cube of each integer starting from 1:

$1^3 = 1$
$2^3 = 8$
$3^3 = 27$
$4^3 = 64$
$5^3 = 125$
$6^3 = 216$
$7^3 = 343$
$8^3 = 512$
$9^3 = 729$
$10^3 = 1000$

The last integer $n$ for which $n^3 \leq 1000$ is $10$.

List the cubic roots The cubic roots of the cubes we calculated are simply the integers $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$.

Therefore, the cubic roots corresponding to whole cubes within the range of 1 to 1000 are:

1 (cubic root of $1$)
2 (cubic root of $8$)
3 (cubic root of $27$)
4 (cubic root of $64$)
5 (cubic root of $125$)
6 (cubic root of $216$)
7 (cubic root of $343$)
8 (cubic root of $512$)
9 (cubic root of $729$)
10 (cubic root of $1000$)

The cubic roots of integers within the range of 1 to 1000 are: $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$.

More Information

Each of these numbers is the cubic root of a corresponding whole number that is also a cube of an integer. The largest cube in the specified range is $1000$, which is the cube of $10$.

Tips

Confusing the cubes with their roots. It’s important to identify the cubic roots rather than the cubed numbers themselves.
Failing to check that the cubes actually fall within the range of 1 to 1000.

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