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 the ratio of their volumes is 1:8, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Volume Formula**:
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 Ratio of Volumes**:
Let the volumes of the two spheres be \( V_1 \) and \( V_2 \). According to the problem, we have:
\[ \frac{V_1}{V_2} = \frac{1}{8} \]

3. **Substitute the Volume Formula**:
Using the volume formula, we can express the volumes in terms of their radii \( R_1 \) and \( R_2 \):
\[ \frac{V_1}{V_2} = \frac{\frac{4}{3} \pi R_1^3}{\frac{4}{3} \pi R_2^3} \]

4. **Simplify the Expression**:
The \( \frac{4}{3} \pi \) terms cancel out:
\[ \frac{R_1^3}{R_2^3} = \frac{1}{8} \]

5. **Relate the Cubes to the Ratio**:
We can rewrite \( \frac{1}{8} \) as \( \frac{1^3}{2^3} \):
\[ \frac{R_1^3}{R_2^3} = \frac{1^3}{2^3} \]

6. **Take the Cube Root**:
Taking the cube root of both sides gives us the ratio of the radii:
\[ \frac{R_1}{R_2} = \frac{1}{2} \]

7. **Final Ratio of Radii**:
Thus, the ratio of the radii \( R_1 : R_2 \) is:
\[ R_1 : R_2 = 1 : 2 \]

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