Calculate the nuclear radii of `._(56)Ba^(140)` and `._(8)O^(17)`. Given `R_(0) = 1.5 fm`.

Answer

To calculate the nuclear radii of \( _{56}^{140}\text{Ba} \) (Barium) and \( _{8}^{17}\text{O} \) (Oxygen), we will use the formula for nuclear radius:

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

where:
- \( R \) is the nuclear radius,
- \( R_0 \) is a constant (given as \( 1.5 \, \text{fm} \)),
- \( A \) is the mass number of the nucleus.

### Step 1: Calculate the nuclear radius of Barium \( _{56}^{140}\text{Ba} \)

1. Identify the mass number \( A \) for Barium:
\[ A = 140 \]

2. Substitute \( R_0 \) and \( A \) into the formula:
\[ R_1 = R_0 A^{1/3} = 1.5 \, \text{fm} \times (140)^{1/3} \]

3. Calculate \( (140)^{1/3} \):
\[ (140)^{1/3} \approx 5.241 \]

4. Now calculate \( R_1 \):
\[ R_1 = 1.5 \, \text{fm} \times 5.241 \approx 7.862 \, \text{fm} \]

### Step 2: Calculate the nuclear radius of Oxygen \( _{8}^{17}\text{O} \)

1. Identify the mass number \( A \) for Oxygen:
\[ A = 17 \]

2. Substitute \( R_0 \) and \( A \) into the formula:
\[ R_2 = R_0 A^{1/3} = 1.5 \, \text{fm} \times (17)^{1/3} \]

3. Calculate \( (17)^{1/3} \):
\[ (17)^{1/3} \approx 2.571 \]

4. Now calculate \( R_2 \):
\[ R_2 = 1.5 \, \text{fm} \times 2.571 \approx 3.857 \, \text{fm} \]

### Final Results:
- The nuclear radius of \( _{56}^{140}\text{Ba} \) is approximately \( 7.86 \, \text{fm} \).
- The nuclear radius of \( _{8}^{17}\text{O} \) is approximately \( 3.86 \, \text{fm} \).

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