The value of `1/(sqrt40+sqrt20+sqrt10-sqrt80)` is equal to

Options

A. `1/70(3sqrt10+2sqrt5)`
B. `1/70(3sqrt10-2sqrt5)`
C. `(-1)/70(3sqrt10+2sqrt5)`
D. `1/70(2sqrt5-3sqrt10)`

Answer

To solve the expression \( \frac{1}{\sqrt{40} + \sqrt{20} + \sqrt{10} - \sqrt{80}} \), we will simplify the terms in the denominator step by step.

### Step 1: Simplify each square root term
1. \( \sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10} \)
2. \( \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \)
3. \( \sqrt{10} = \sqrt{10} \)
4. \( \sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5} \)

### Step 2: Substitute the simplified terms back into the expression
Now, substitute these simplified terms back into the denominator:
\[ \sqrt{40} + \sqrt{20} + \sqrt{10} - \sqrt{80} = 2\sqrt{10} + 2\sqrt{5} + \sqrt{10} - 4\sqrt{5} \]

### Step 3: Combine like terms
Combine the terms:
\[ (2\sqrt{10} + \sqrt{10}) + (2\sqrt{5} - 4\sqrt{5}) = 3\sqrt{10} - 2\sqrt{5} \]

### Step 4: Rewrite the expression
Now we can rewrite the expression:
\[ \frac{1}{3\sqrt{10} - 2\sqrt{5}} \]

### Step 5: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{1 \cdot (3\sqrt{10} + 2\sqrt{5})}{(3\sqrt{10} - 2\sqrt{5})(3\sqrt{10} + 2\sqrt{5})} \]

### Step 6: Calculate the denominator using the difference of squares
Using the formula \( (a - b)(a + b) = a^2 - b^2 \):
\[ (3\sqrt{10})^2 - (2\sqrt{5})^2 = 9 \cdot 10 - 4 \cdot 5 = 90 - 20 = 70 \]

### Step 7: Write the final expression
Now we can write the final expression:
\[ \frac{3\sqrt{10} + 2\sqrt{5}}{70} \]

### Final Answer
Thus, the value of \( \frac{1}{\sqrt{40} + \sqrt{20} + \sqrt{10} - \sqrt{80}} \) is:
\[ \frac{3\sqrt{10} + 2\sqrt{5}}{70} \]