What is the simplified value of `(sqrt5 + sqrt4)/(sqrt5- sqrt4) + (sqrt5 - sqrt4)/(sqrt5 + sqrt4)` ?

Options

A. 9
B. 10
C. 18
D. 20

Answer

To simplify the expression \((\sqrt{5} + \sqrt{4})/(\sqrt{5} - \sqrt{4}) + (\sqrt{5} - \sqrt{4})/(\sqrt{5} + \sqrt{4})\), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ \frac{\sqrt{5} + \sqrt{4}}{\sqrt{5} - \sqrt{4}} + \frac{\sqrt{5} - \sqrt{4}}{\sqrt{5} + \sqrt{4}} \]

### Step 2: Find a Common Denominator
The common denominator for the two fractions is \((\sqrt{5} - \sqrt{4})(\sqrt{5} + \sqrt{4})\). Thus, we rewrite the expression:
\[ \frac{(\sqrt{5} + \sqrt{4})^2 + (\sqrt{5} - \sqrt{4})^2}{(\sqrt{5} - \sqrt{4})(\sqrt{5} + \sqrt{4})} \]

### Step 3: Expand the Numerator
Now we expand the numerator:
\[ (\sqrt{5} + \sqrt{4})^2 = (\sqrt{5})^2 + 2\sqrt{5}\sqrt{4} + (\sqrt{4})^2 = 5 + 2 \cdot \sqrt{5} \cdot 2 + 4 = 5 + 4 + 4\sqrt{5} = 9 + 4\sqrt{5} \]
\[ (\sqrt{5} - \sqrt{4})^2 = (\sqrt{5})^2 - 2\sqrt{5}\sqrt{4} + (\sqrt{4})^2 = 5 - 4 + 4\sqrt{5} = 1 - 4\sqrt{5} \]

### Step 4: Combine the Expanded Numerators
Now we combine both parts:
\[ (9 + 4\sqrt{5}) + (1 - 4\sqrt{5}) = 10 + 0 = 10 \]

### Step 5: Simplify the Denominator
Now we simplify the denominator:
\[ (\sqrt{5} - \sqrt{4})(\sqrt{5} + \sqrt{4}) = (\sqrt{5})^2 - (\sqrt{4})^2 = 5 - 4 = 1 \]

### Step 6: Final Expression
Putting it all together, we have:
\[ \frac{10}{1} = 10 \]

### Final Answer
Thus, the simplified value of the expression is:
\[ \boxed{10} \]