How can we simplify this expression involving square roots?

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Discussion Overview

The discussion revolves around simplifying the expression involving square roots: $$4\sqrt{n^{2}}+\sqrt{m^{2}n-\sqrt{4n^{2}}}-\sqrt{mn^{2}}$$. Participants explore various approaches to simplification, including the application of properties of square roots.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest starting with simplifying $\sqrt{n^2}$, questioning what that evaluates to.
  • One participant proposes that if $n$ is non-negative, then $\sqrt{n^2} = n$, while noting that otherwise it should be expressed as $|n|$.
  • Another participant emphasizes observing and simplifying rather than distributing, mentioning that $4\sqrt{n^2} = 4n$ and recalling properties such as $\sqrt{ab} = \sqrt{a}\cdot\sqrt{b}$.

Areas of Agreement / Disagreement

Participants generally agree on the need to simplify the expression, but there is no consensus on the specific methods to apply or the implications of the assumptions regarding the non-negativity of $n$.

Contextual Notes

There are limitations regarding the assumptions about the values of $n$ and $m$, particularly concerning the non-negativity of $n$ and how it affects the simplification of $\sqrt{n^2}$.

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?
 
Last edited:
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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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