MHB How can we simplify this expression involving square roots?

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The expression $$4\sqrt{n^{2}}+\sqrt{m^{2}n-\sqrt{4n^{2}}}-\sqrt{mn^{2}}$$ can be simplified by recognizing that $\sqrt{n^2} = n$ and $\sqrt{4} = 2$. This leads to the simplification of the first term to $4n$. The discussion emphasizes the importance of using properties of square roots, such as $\sqrt{ab} = \sqrt{a}\cdot\sqrt{b}$. Additionally, it is noted that if $n$ is non-negative, $\sqrt{n^2}$ simplifies directly to $n$, while in general, it should be expressed as $|n|$. Understanding these simplifications is key to tackling the expression effectively.
Zoey323
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$$4\sqrt{n^{2}}+\sqrt{m^{2}n-\sqrt{4n^{2}}}-\sqrt{mn^{2}}$$
I don't even know where to start, I know the teacher said to distribute but distibute what?
 
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Zoey323 said:
$$4\sqrt{n^{2}}+\sqrt{m^{2}n-\sqrt{4n^{2}}}-\sqrt{mn^{2}}$$
I don't even know where to start, I know the teacher said to distribute but distibute what?
Well, you could start by simplifying [math]\sqrt{n^2}[/math]. What is that?

-Dan
 
Zoey323 said:
$$4\sqrt{n^{2}}+\sqrt{m^{2}n-\sqrt{4n^{2}}}-\sqrt{mn^{2}}$$
I don't even know where to start, I know the teacher said to distribute but distibute what?

Not necessarily distribute but observe and simplify.

Use facts like

$\sqrt(n^2) = n$ and facts like $\sqrt(4) = 2$

To start you off, notice,

$4\sqrt(n^2) = 4n$

Also remember facts like

$\sqrt(ab) = \sqrt(a)\cdot\sqrt(b)$
 
If $n$ is known to be non-negative, then we can write:

$$\sqrt{n^2}=n$$

Otherwise, we should write by definition:

$$\sqrt{n^2}=|n|$$
 
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