If the ratio of the volumes of two spheres is 1 : 8, then find the ratio of their radii.

Answer

To solve the problem of finding the ratio of the radii of two spheres given that the ratio of their volumes is 1:8, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Volume Formula for a Sphere**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

2. **Set Up the Volume Ratio**:
Let the radius of the first sphere be \( r_1 \) and the radius of the second sphere be \( r_2 \). According to the problem, the ratio of their volumes is:
\[ \frac{V_1}{V_2} = \frac{1}{8} \]
Substituting the volume formula, we have:
\[ \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \frac{1}{8} \]

3. **Simplify the Equation**:
The \( \frac{4}{3} \pi \) terms cancel out, so we simplify to:
\[ \frac{r_1^3}{r_2^3} = \frac{1}{8} \]

4. **Take the Cube Root of Both Sides**:
To find the ratio of the radii, we take the cube root of both sides:
\[ \frac{r_1}{r_2} = \frac{1}{2} \]

5. **Express the Ratio of Radii**:
This means that the ratio of the radii \( r_1 \) to \( r_2 \) is:
\[ r_1 : r_2 = 1 : 2 \]

### Final Answer:
The ratio of the radii of the two spheres is \( 1 : 2 \).