Compare the radii of two nuclei with mass number 8 and 125 respectively.

Answer

To compare the radii of two nuclei with mass numbers 8 and 125, we can use the formula for the radius of a nucleus, which is given by:

\[ R = R_0 \cdot A^{1/3} \]

where:
- \( R \) is the radius of the nucleus,
- \( R_0 \) is a constant (approximately \( 1.2 \, \text{fm} \) for nuclear radii),
- \( A \) is the mass number of the nucleus.

### Step-by-Step Solution:

1. **Identify Mass Numbers:**
- Let \( A_1 = 8 \) (for the first nucleus)
- Let \( A_2 = 125 \) (for the second nucleus)

2. **Write the Radius Formula for Both Nuclei:**
- The radius of the first nucleus (\( R_1 \)) is:
\[ R_1 = R_0 \cdot A_1^{1/3} = R_0 \cdot 8^{1/3} \]
- The radius of the second nucleus (\( R_2 \)) is:
\[ R_2 = R_0 \cdot A_2^{1/3} = R_0 \cdot 125^{1/3} \]

3. **Set Up the Ratio of Radii:**
- To compare the two radii, we take the ratio \( \frac{R_1}{R_2} \):
\[ \frac{R_1}{R_2} = \frac{R_0 \cdot 8^{1/3}}{R_0 \cdot 125^{1/3}} \]

4. **Cancel Out the Constant \( R_0 \):**
- Since \( R_0 \) is common in both the numerator and denominator, it cancels out:
\[ \frac{R_1}{R_2} = \frac{8^{1/3}}{125^{1/3}} \]

5. **Simplify the Ratio:**
- The ratio can be simplified to:
\[ \frac{R_1}{R_2} = \left(\frac{8}{125}\right)^{1/3} \]

6. **Calculate the Values:**
- We know that \( 8 = 2^3 \) and \( 125 = 5^3 \), so:
\[ \frac{8}{125} = \frac{2^3}{5^3} = \left(\frac{2}{5}\right)^3 \]
- Therefore:
\[ \frac{R_1}{R_2} = \left(\frac{2}{5}\right)^{1} = \frac{2}{5} \]

### Final Result:
The ratio of the radii of the two nuclei with mass numbers 8 and 125 is:
\[ \frac{R_1}{R_2} = \frac{2}{5} \]