Suppose a, b are positive real numbers such that `asqrt(a) + b sqrt(b) = 183, asqrt(b) + bsqrt(a) = 182`. Find `9/5(a+b)`.

Answer

To solve the equations given:

1. **Equations**:
- \( a \sqrt{a} + b \sqrt{b} = 183 \) (Equation 1)
- \( a \sqrt{b} + b \sqrt{a} = 182 \) (Equation 2)

2. **Rewriting the equations**:
- Let \( x = \sqrt{a} \) and \( y = \sqrt{b} \). Then, we can rewrite the equations as:
- \( x^3 + y^3 = 183 \) (since \( a = x^2 \) and \( b = y^2 \))
- \( xy(x + y) = 182 \)

3. **Using the identity for cubes**:
- We know that \( x^3 + y^3 = (x + y)(x^2 - xy + y^2) \).
- We can express \( x^2 - xy + y^2 \) as \( (x + y)^2 - 3xy \).

4. **Let \( s = x + y \) and \( p = xy \)**:
- Then, we can rewrite the first equation as:
\[ (x + y)((x + y)^2 - 3xy) = 183 \]
\[ s(s^2 - 3p) = 183 \quad \text{(Equation 3)} \]

5. **From the second equation**:
- We have:
\[ p \cdot s = 182 \quad \text{(Equation 4)} \]

6. **Substituting Equation 4 into Equation 3**:
- From Equation 4, we can express \( p \) as \( p = \frac{182}{s} \).
- Substitute this into Equation 3:
\[ s \left(s^2 - 3 \cdot \frac{182}{s}\right) = 183 \]
\[ s^3 - 546 = 183 \]
\[ s^3 = 729 \]
\[ s = 9 \]

7. **Finding \( p \)**:
- Substitute \( s = 9 \) back into Equation 4:
\[ p \cdot 9 = 182 \implies p = \frac{182}{9} \]

8. **Finding \( a + b \)**:
- We know \( a + b = (x + y)^2 = s^2 = 9^2 = 81 \).

9. **Finding \( \frac{9}{5}(a + b) \)**:
- Now we need to find \( \frac{9}{5}(a + b) \):
\[ \frac{9}{5}(a + b) = \frac{9}{5} \cdot 81 = \frac{729}{5} = 145.8 \]

Thus, the final answer is:
\[ \frac{9}{5}(a + b) = 145.8 \]