If `sqrt(5)=2.236` and `sqrt(10)=3.162` , then the value of `(15)/(sqrt(10)+sqrt(20)+sqrt(40)-sqrt(5)-sqrt(80))` is

Options

A. `5.398`
B. `4.398`
C. `3.398`
D. `6.398`

Answer

To solve the expression \(\frac{15}{\sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80}}\), we will simplify the denominator step by step.

### Step 1: Rewrite the square roots in the denominator
We know that:
- \(\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}\)
- \(\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}\)
- \(\sqrt{80} = \sqrt{16 \cdot 5} = 4\sqrt{5}\)

Now, substitute these values into the denominator:
\[ \sqrt{10} + \sqrt{20} + \sqrt{40} - \sqrt{5} - \sqrt{80} = \sqrt{10} + 2\sqrt{5} + 2\sqrt{10} - \sqrt{5} - 4\sqrt{5} \]

### Step 2: Combine like terms in the denominator
Now, combine the terms:
\[ \sqrt{10} + 2\sqrt{10} + 2\sqrt{5} - \sqrt{5} - 4\sqrt{5} = 3\sqrt{10} - 3\sqrt{5} \]

### Step 3: Factor out the common term
The denominator can be factored:
\[ 3(\sqrt{10} - \sqrt{5}) \]

### Step 4: Substitute the simplified denominator back into the expression
Now, substitute this back into the original expression:
\[ \frac{15}{3(\sqrt{10} - \sqrt{5})} \]

### Step 5: Simplify the fraction
Now simplify the fraction:
\[ \frac{15}{3(\sqrt{10} - \sqrt{5})} = \frac{15}{3} \cdot \frac{1}{\sqrt{10} - \sqrt{5}} = 5 \cdot \frac{1}{\sqrt{10} - \sqrt{5}} = \frac{5}{\sqrt{10} - \sqrt{5}} \]

### Step 6: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{10} + \sqrt{5}\):
\[ \frac{5(\sqrt{10} + \sqrt{5})}{(\sqrt{10} - \sqrt{5})(\sqrt{10} + \sqrt{5})} = \frac{5(\sqrt{10} + \sqrt{5})}{10 - 5} = \frac{5(\sqrt{10} + \sqrt{5})}{5} \]

### Step 7: Final simplification
This simplifies to:
\[ \sqrt{10} + \sqrt{5} \]

### Step 8: Substitute the values of \(\sqrt{10}\) and \(\sqrt{5}\)
Now substitute the given values:
\(\sqrt{10} = 3.162\) and \(\sqrt{5} = 2.236\):
\[ \sqrt{10} + \sqrt{5} = 3.162 + 2.236 = 5.398 \]

Thus, the final answer is:
\[ \sqrt{10} + \sqrt{5} = 5.398 \]