"rationalize" the denominator rather than the numerator

  • Thread starter powp
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In summary, to simplify expressions with radicals, rationalizing the denominator is typically preferred. In the case of \frac{\sqrt{x}+1}{\sqrt{x}-1}, we can rationalize the denominator to get \frac{(1+\sqrt{x})^2}{x-1}. Additionally, for the expression \sqrt{x}(\sqrt{x}+1)(2\sqrt{x}-1), we can simplify further to get 2x \sqrt{x}+x- \sqrt{x}.
  • #1
powp
91
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Hello All

I get to thrown off when there is a question with radicals . Can you please help

Simplify the following

[tex]\frac{\sqrt{x} + 1}{\sqrt{x} - 1}[/tex]


and Preform the indicated operations and simplfy


[tex]\sqrt{x}(\sqrt{x} + 1)(2\sqrt{x}-1)[/tex]

Thanks

P
 
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  • #2
the first:

[tex]\frac{\sqrt{x} + 1}{\sqrt{x} - 1}[/tex]

[tex]\frac{(\sqrt{x} + 1)(\sqrt{x} - 1)}{(\sqrt{x} - 1)(\sqrt{x} - 1)}[/tex]

[tex]\frac{x-1}{x -2 \sqrt{x} +1}[/tex]

the second:

[tex]\sqrt{x}(\sqrt{x} + 1)(2\sqrt{x}-1)[/tex]

[tex]\sqrt{x}(2x+ \sqrt{x}-1)[/tex]

[tex]2x \sqrt{x}+x- \sqrt{x}[/tex]
 
  • #3
I would have been inclined to "rationalize" the denominator rather than the numerator:
[tex]\frac{\sqrt{x}+1}{\sqrt{x}-1}= \frac{(\sqrt{x}+1)(\sqrt{x}+1)}{(\sqrt{x}-1})(\sqrt{x}+1)}= \frac{x+ 2\sqrt{x}+1}{x-1}[/tex]
 
  • #4
thanks that's the answers I got but was not sure

When would you "rationalize" the denominator over the numerator or the other way around.
 
  • #5
Generally simplified form means always "rationalizing" the denominator. Also you can simplify a little bit more. [tex] \frac{x+ 2\sqrt{x}+1}{x-1} = \frac{ (1+\sqrt{x})^2}{x-1}[/tex]
 

Related to "rationalize" the denominator rather than the numerator

1. Why is it important to rationalize the denominator rather than the numerator?

Rationalizing the denominator makes the expression easier to work with and simplifies the final answer. It also eliminates any potential for division by zero, which is undefined.

2. How do you rationalize the denominator?

To rationalize the denominator, you multiply the numerator and denominator by a suitable expression that will eliminate any radicals or imaginary numbers in the denominator. This expression is typically the conjugate of the denominator.

3. Can you provide an example of rationalizing the denominator?

Sure, let's say we have the expression (2 + √3)/√5. To rationalize the denominator, we would multiply the expression by (√5/√5), which is the conjugate of √5. This would give us (2√5 + √15)/5.

4. Is it always necessary to rationalize the denominator?

No, it is not always necessary. In some cases, rationalizing the denominator may not make the expression any simpler or easier to work with. However, in certain situations, it is important to rationalize the denominator to avoid undefined values or to simplify the expression.

5. Are there any other benefits to rationalizing the denominator?

Yes, in addition to making the expression easier to work with, rationalizing the denominator can also help with finding equivalent fractions and comparing values between different expressions.

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