Evaluate `15/( sqrt10+sqrt20+sqrt40-sqrt5-sqrt80)` is being given that `sqrt5=2.236` and `sqrt10=3.162`

Explanation

To evaluate the expression \( \frac{15}{\sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80}} \), we will follow these steps:

### Step 1: Substitute the known values
We are given:
- \( \sqrt{5} = 2.236 \)
- \( \sqrt{10} = 3.162 \)

### Step 2: Calculate \( \sqrt{20} \)
We can express \( \sqrt{20} \) as:
\[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \]
Now substituting the value of \( \sqrt{5} \):
\[ \sqrt{20} = 2 \times 2.236 = 4.472 \]

### Step 3: Calculate \( \sqrt{40} \)
We can express \( \sqrt{40} \) as:
\[ \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10} \]
Now substituting the value of \( \sqrt{10} \):
\[ \sqrt{40} = 2 \times 3.162 = 6.324 \]

### Step 4: Calculate \( \sqrt{80} \)
We can express \( \sqrt{80} \) as:
\[ \sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5} \]
Now substituting the value of \( \sqrt{5} \):
\[ \sqrt{80} = 4 \times 2.236 = 8.944 \]

### Step 5: Substitute all values into the denominator
Now we can substitute all calculated values into the denominator:
\[ \text{Denominator} = \sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80} \]
Substituting the known values:
\[ = 3.162 + 4.472 + 6.324 - 2.236 - 8.944 \]

### Step 6: Calculate the denominator
First, calculate the sum of the positive terms:
\[ 3.162 + 4.472 + 6.324 = 13.958 \]
Now calculate the sum of the negative terms:
\[ 2.236 + 8.944 = 11.18 \]
Now subtract:
\[ 13.958 - 11.18 = 2.778 \]

### Step 7: Final calculation
Now we can substitute this back into the original expression:
\[ \frac{15}{2.778} \]
Calculating this gives:
\[ \approx 5.3995 \]

### Final Answer
Thus, the value of the expression is approximately \( 5.3995 \).