Find the value of `(1.02)^(3//2) - (0.98)^(3//2)` correct to 6 decimals .

Answer

To find the value of \( (1.02)^{\frac{3}{2}} - (0.98)^{\frac{3}{2}} \) correct to six decimal places, we can use the binomial theorem to expand both terms.

### Step-by-Step Solution:

1. **Rewrite the terms**:
\[ (1.02)^{\frac{3}{2}} = (1 + 0.02)^{\frac{3}{2}} \quad \text{and} \quad (0.98)^{\frac{3}{2}} = (1 - 0.02)^{\frac{3}{2}} \]

2. **Use the Binomial Expansion**:
The binomial expansion for \( (1 + x)^n \) is given by:
\[ (1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \ldots \]
For \( (1 + 0.02)^{\frac{3}{2}} \):
- Here, \( n = \frac{3}{2} \) and \( x = 0.02 \).
\[ (1 + 0.02)^{\frac{3}{2}} \approx 1 + \frac{3}{2}(0.02) + \frac{\frac{3}{2}(\frac{3}{2}-1)}{2}(0.02)^2 + \frac{\frac{3}{2}(\frac{3}{2}-1)(\frac{3}{2}-2)}{6}(0.02)^3 \]

3. **Calculate the first few terms**:
- First term: \( 1 \)
- Second term:
\[ \frac{3}{2}(0.02) = 0.03 \]
- Third term:
\[ \frac{\frac{3}{2} \cdot \frac{1}{2}}{2}(0.02)^2 = \frac{3/4}{2}(0.0004) = \frac{3}{8}(0.0004) = 0.00015 \]
- Fourth term:
\[ \frac{\frac{3}{2} \cdot \frac{1}{2} \cdot -\frac{1}{2}}{6}(0.02)^3 = \frac{-3/8}{6}(0.000008) = -0.000004 \]

4. **Combine the terms for \( (1.02)^{\frac{3}{2}} \)**:
\[ (1.02)^{\frac{3}{2}} \approx 1 + 0.03 + 0.00015 - 0.000004 \approx 1.030146 \]

5. **Now for \( (0.98)^{\frac{3}{2}} \)**:
Using the binomial expansion:
\[ (1 - 0.02)^{\frac{3}{2}} \approx 1 - \frac{3}{2}(0.02) + \frac{\frac{3}{2}(\frac{1}{2})}{2}(0.02)^2 - \frac{\frac{3}{2}(\frac{1}{2})(-\frac{1}{2})}{6}(0.02)^3 \]
- First term: \( 1 \)
- Second term:
\[ -\frac{3}{2}(0.02) = -0.03 \]
- Third term:
\[ \frac{3/4}{2}(0.0004) = 0.00015 \]
- Fourth term:
\[ \frac{-3/8}{6}(0.000008) = -0.000004 \]

6. **Combine the terms for \( (0.98)^{\frac{3}{2}} \)**:
\[ (0.98)^{\frac{3}{2}} \approx 1 - 0.03 + 0.00015 - 0.000004 \approx 0.970146 \]

7. **Final Calculation**:
\[ (1.02)^{\frac{3}{2}} - (0.98)^{\frac{3}{2}} \approx 1.030146 - 0.970146 = 0.060000 \]

### Final Answer:
The value of \( (1.02)^{\frac{3}{2}} - (0.98)^{\frac{3}{2}} \) correct to six decimal places is:
\[ \boxed{0.060000} \]