\frac{3\sqrt{2}-\sqrt{3}-\sqrt{6}+\sqrt{16}}{\sqrt{2}-\sqrt{3}+\sqrt{4}}

Let's simplify the expression step by step:

\[
\frac{3\sqrt{2}-\sqrt{3}-\sqrt{6}+\sqrt{16}}{\sqrt{2}-\sqrt{3}+\sqrt{4}}
\]

### Step 1: Simplify square roots where possible.
- \(\sqrt{16} = 4\)
- \(\sqrt{4} = 2\)

Substituting these values into the expression:

\[
\frac{3\sqrt{2} - \sqrt{3} - \sqrt{6} + 4}{\sqrt{2} - \sqrt{3} + 2}
\]

### Step 2: Attempt to simplify the numerator and denominator.

The numerator is:
\[
3\sqrt{2} - \sqrt{3} - \sqrt{6} + 4
\]

The denominator is:
\[
\sqrt{2} - \sqrt{3} + 2
\]

Unfortunately, we can't factor these expressions further without further context, but we can simplify them numerically if needed. To do that, let's approximate the square roots:

- \(\sqrt{2} \approx 1.414\)
- \(\sqrt{3} \approx 1.732\)
- \(\sqrt{6} \approx 2.449\)

### Step 3: Substitute approximations.

Numerator:
\[
3(1.414) - 1.732 - 2.449 + 4 \approx 4.242 - 1.732 - 2.449 + 4 = 4.061
\]

Denominator:
\[
1.414 - 1.732 + 2 \approx 1.414 - 1.732 + 2 = 1.682
\]

### Step 4: Divide the numerator by the denominator.
\[
\frac{4.061}{1.682} \approx 2.414
\]

Thus, the value of the expression is approximately:
\[
\boxed{2.414}
\]

---

Would you like me to go into more detail or explore a different aspect of this problem? Here's a list of related questions you might find interesting:

1. How can we simplify expressions involving square roots algebraically without approximating?
2. What is the concept of rationalizing the denominator and how can it be applied here?
3. How does the distribution of square roots work in multiplication or division?
4. Can we simplify this expression further using trigonometric identities?
5. What are some methods for approximating square roots more accurately?

**Tip:** When dealing with square roots in expressions, always look for opportunities to simplify terms, either by combining like terms or rationalizing the denominator if necessary.

frac{3sqrt{2}-sqrt{3}-sqrt{6}+sqrt{16}}{sqrt{2}-sqrt{3}+sqrt{4}}

Learn how to simplify the expression 3sqrt(2) - sqrt(3) - sqrt(6) + sqrt(16) divided by sqrt(2) - sqrt(3) + sqrt(4) step-by-step. This problem covers square root simplification, numerical approximation techniques, and algebraic expressions. Ideal for students in Grades 9-11.

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