Simplify: $-\sqrt{2}-4 \sqrt{4}-3 \sqrt{2}$
We need to simplify the expression \(-\sqrt{2} - 4\sqrt{4} - 3\sqrt{2}\). The expression involves terms with \(\sqrt{2}\) and a term involving \(\sqrt{4}\). We will group the like terms and simplify the square roots where possible.
First, observe the term \(4\sqrt{4}\). Since \(4\) is a perfect square, we know that \(\sqrt{4} = 2\). Therefore, we can simplify this term:
\[4\sqrt{4} = 4 \cdot 2 = 8.\]
Now, substitute this back into the original expression:
\[-\sqrt{2} - 8 - 3\sqrt{2}.\]
Next, we group the terms involving \(\sqrt{2}\):
\[(-\sqrt{2}) + (-3\sqrt{2}) = -1\sqrt{2} - 3\sqrt{2} = (-1 - 3)\sqrt{2} = -4\sqrt{2}.\]
Now, combine this with the constant term:
\[-4\sqrt{2} - 8.\]
The expression simplifies to \(-4\sqrt{2} - 8\), which matches one of the provided options. Thus, the correct simplified form of the expression is \(-4\sqrt{2} - 8\).
Simplifying expressions like \(-\sqrt{2} - 4\sqrt{4} - 3\sqrt{2}\) is essential in algebra when dealing with radicals. In real life, this is similar to bookkeeping where you track different types of expenses. For example, if you have two different types of debt (like \(\sqrt{2}\) units of debt and a flat debt), you need to aggregate identical categories to find your total balance. This helps in managing budgets or calculating net results in physical dimensions where units must be combined correctly.