MHB How can we simplify this expression involving square roots?

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SUMMARY

The discussion focuses on simplifying the expression $$4\sqrt{n^{2}}+\sqrt{m^{2}n-\sqrt{4n^{2}}}-\sqrt{mn^{2}}$$. Key simplifications include recognizing that $$\sqrt{n^2} = n$$ and $$\sqrt{4} = 2$$, leading to the transformation of $$4\sqrt{n^2}$$ into $$4n$$. Participants emphasize the importance of understanding properties of square roots, such as $$\sqrt{ab} = \sqrt{a}\cdot\sqrt{b}$$, particularly when $$n$$ is non-negative.

PREREQUISITES
  • Understanding of square root properties, including $$\sqrt{a}$$ and $$\sqrt{ab}$$.
  • Familiarity with algebraic expressions and simplification techniques.
  • Knowledge of absolute values, specifically $$|n|$$ when dealing with square roots.
  • Basic arithmetic operations involving constants and variables.
NEXT STEPS
  • Study the properties of square roots in depth, focusing on $$\sqrt{a}$$ and $$\sqrt{ab}$$.
  • Practice simplifying algebraic expressions involving square roots.
  • Learn about the implications of absolute values in mathematical expressions.
  • Explore advanced algebra techniques for simplifying complex expressions.
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Students studying algebra, mathematics educators, and anyone looking to enhance their skills in simplifying expressions involving square roots.

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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