Cubes and cube roots

Chapter: Cubes and Roots

1. Find the smallest number by which the following numbers should be divided to obtain a perfect cube.

   1. $13718$
   2. $28672$

   Solution:

   1. The smallest number to divide $13718$ to obtain a perfect cube is $2$.
   2. The smallest number to divide $28672$ to obtain a perfect cube is $1$ (since $28672$ is already a perfect cube).

2. Find the volume of a cubical box whose surface area is $486 cm^2$.

   Solution:

   The surface area of a cube is given by $6a^2$, where $a$ is the length of one side. Hence, $a = \sqrt{\frac{Surface Area}{6}} = \sqrt{\frac{486}{6}} = 9 cm$. The volume of a cube is $a^3 = 9^3 = 729 cm^3$.

3. Find the cube of $-5 \frac{1}{7}$.

   Solution:

   $-5 \frac{1}{7} = -\frac{36}{7}$, so its cube is $\left(-\frac{36}{7}\right)^3 = -\frac{46656}{343}$.

4. Divide the number $8748$ by the smallest number so that the quotient is a perfect cube. Also find the cube root of the quotient.

   Solution:

   The smallest number to divide $8748$ to obtain a perfect cube is $3$. So, the quotient is $\frac{8748}{3} = 2916$, which is a perfect cube. The cube root of $2916$ is $14$.

5. Three numbers are in the ratio $3:4:5$. If their product is $480$, find the numbers.

   Solution:

   If the three numbers are in the ratio $3:4:5$, let's denote the numbers as $3x, 4x,$ and $5x$. Therefore, $3x \cdot 4x \cdot 5x = 480$ which gives $x^3 = \frac{480}{60} = 8$. So, $x = \sqrt[3]{8} = 2$. Thus, the three numbers are $3x = 6, 4x = 8,$ and $5x = 10$.