MHB Square Root Rules for Fractions: x∈[3,∞)

AI Thread Summary
The discussion centers on the expression \(\sqrt{\frac{x-3}{x}} = \frac{\sqrt{x-3}}{\sqrt{x}}\), which is valid only for \(x \in [3, \infty)\). Participants emphasize the importance of understanding the domain restrictions, noting that the expression does not hold for all \(x\), particularly when \(x < 3\) or \(x < 0\). They suggest that teaching this concept to high school students should involve clarifying these domain issues to avoid confusion with imaginary numbers. The conversation highlights the need to split the analysis into cases based on the value of \(x\) to maintain clarity and accuracy. Overall, the focus is on ensuring students grasp the conditions under which the square root rules apply.
Amer
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\sqrt{\frac{x-3}{x}} = \frac{\sqrt{x-3}}{\sqrt{x}}

that is not true for all x, it is true for x\in [3,\infty)
I want to teach my students that the exponents distribute over fractions unless we have a case like that square root or any even root.
what do you think ?
 
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Re: about square root

Generally for the case

$$\sqrt{\frac{a}{b}} $$ we require that

$$\frac{a}{b}\geq 0 $$ which means either

  • a $\geq$ 0,b>0
  • $a\leq 0$,b<0

For the first case we can state $$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}} $$
 
Re: about square root

What made me ask this question the website wolfarmalpha plot the function
f(x) = \frac{\sqrt{x-3}}{\sqrt{x}}
View attachment 1433

although
f(-1) = \frac{\sqrt{-4}}{\sqrt{-2}} which is not a real number
the domain of the function is [3,\infty)
f should start from x=3 to infinity, Am I right ?
 

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Re: about square root

Wolfram gives the plot of the function in the complex plane , we have to add the condition that $x \geq 3$ to only focus on real part.

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Amer said:
f(-1) = \frac{\sqrt{-4}}{\sqrt{-2}} which is not a real number

Unfortunately this is a real number f(-1) = \frac{\sqrt{-4}}{\sqrt{-2}} =\frac{2i}{\sqrt{2} i}=\sqrt{2}
 
Re: about square root

Amer said:
\sqrt{\frac{x-3}{x}} = \frac{\sqrt{x-3}}{\sqrt{x}}
ZaidAlyafey's response shows that there is, in fact, a distinction between [math]\sqrt{\frac{x - 3}{x}}[/math] and [math]\frac{\sqrt{x - 3}}{\sqrt{x}}[/math]. What level are your students at? It might be better to sweep this particular kind of example under the rug.

-Dan
 
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I see this as a domain issue. If we consider the inequality:

$$\frac{x-3}{x}\ge0$$

we find $x$ in:

$$(-\infty,0)\,\cup\,[3,\infty)$$

And over this domain, we may state:

$$\sqrt{\frac{x-3}{x}}=\frac{\sqrt{x-3}}{\sqrt{x}}$$
 
Re: about square root

topsquark said:
ZaidAlyafey's response shows that there is, in fact, a distinction between [math]\sqrt{\frac{x - 3}{3}}[/math] and [math]\frac{\sqrt{x - 3}}{\sqrt{x}}[/math]. What level are your students at? It might be better to sweep this particular kind of example under the rug.

-Dan
They are high school students 11th class

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MarkFL said:
I see this as a domain issue. If we consider the inequality:

$$\frac{x-3}{x}\ge0$$

we find $x$ in:

$$(-\infty,0)\,\cup\,[3,\infty)$$

And over this domain, we may state:

$$\sqrt{\frac{x-3}{x}}=\frac{\sqrt{x-3}}{\sqrt{x}}$$

I think we should split it into two parts if x>=3
\sqrt{\frac{x-3}{x}} = \frac{\sqrt{x-3}}{\sqrt{x}}

if x< 0
\sqrt{\frac{x-3}{x}} = \frac{\sqrt{3-x}}{\sqrt{-x}}

right ?
 
Re: about square root

Amer said:
...
I think we should split it into two parts if x>=3
\sqrt{\frac{x-3}{x}} = \frac{\sqrt{x-3}}{\sqrt{x}}

if x< 0
\sqrt{\frac{x-3}{x}} = \frac{\sqrt{3-x}}{\sqrt{-x}}

right ?

That would indeed be a better approach, as this way there are no imaginary factors to divide out. (Yes).
 

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