The simplified value of `(sqrt(6)+sqrt(10)-sqrt(21)-sqrt(35))(sqrt(6)-sqrt(10)+sqrt(21)-sqrt(35))` is

Options

A. A) `13`
B. B) `12`
C. C) `11`
D. D) `10`

Answer

To simplify the expression \((\sqrt{6} + \sqrt{10} - \sqrt{21} - \sqrt{35})(\sqrt{6} - \sqrt{10} + \sqrt{21} - \sqrt{35})\), we can use the algebraic identity for the product of two binomials, which is given by:

\[ (a + b)(a - b) = a^2 - b^2 \]

### Step 1: Identify \(a\) and \(b\)
Let:
- \(a = \sqrt{6} + \sqrt{21} - \sqrt{35}\)
- \(b = \sqrt{10}\)

### Step 2: Apply the identity
Now we can rewrite our expression as:

\[ (a + b)(a - b) = a^2 - b^2 \]

### Step 3: Calculate \(a^2\)
First, we calculate \(a^2\):

\[ a^2 = (\sqrt{6} + \sqrt{21} - \sqrt{35})^2 \]

Expanding this:

\[ = (\sqrt{6})^2 + (\sqrt{21})^2 + (\sqrt{35})^2 + 2(\sqrt{6}\sqrt{21}) - 2(\sqrt{6}\sqrt{35}) - 2(\sqrt{21}\sqrt{35}) \]

Calculating each term:

\[ = 6 + 21 + 35 + 2\sqrt{126} - 2\sqrt{210} - 2\sqrt{735} \]

Combining the constants:

\[ = 62 + 2\sqrt{126} - 2\sqrt{210} - 2\sqrt{735} \]

### Step 4: Calculate \(b^2\)
Next, we calculate \(b^2\):

\[ b^2 = (\sqrt{10})^2 = 10 \]

### Step 5: Combine \(a^2\) and \(b^2\)
Now we can substitute back into our expression:

\[ a^2 - b^2 = (62 + 2\sqrt{126} - 2\sqrt{210} - 2\sqrt{735}) - 10 \]

This simplifies to:

\[ = 52 + 2\sqrt{126} - 2\sqrt{210} - 2\sqrt{735} \]

### Step 6: Final simplification
Now we can simplify further, but to find a numerical value, we can calculate the square roots approximately:

- \(\sqrt{126} \approx 11.225\)
- \(\sqrt{210} \approx 14.491\)
- \(\sqrt{735} \approx 27.116\)

Substituting these approximations:

\[ = 52 + 2(11.225) - 2(14.491) - 2(27.116) \]

Calculating:

\[ = 52 + 22.45 - 28.982 - 54.232 \]

Combining these values gives:

\[ = 52 - 60.764 \approx -8.764 \]

Thus, the simplified value of the expression is approximately \(-8.764\).