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I have a doubt. How do you rationalize the following expression? \(\frac{1}{\sqrt[3]{2} + 1}\)

Note by Swapnil Das 2 years, 9 months ago

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2 \times 3

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Let cube root of 2 be x.

Then multiply above and below with (x^2 - x + 1).

You get the denominator as (x^3 + 1) = 3 a rational number.

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Can you explain what is the motivation behind "Multiply with \( x^2 - x + 1 \)?"

For another expression, how do we determine what to do?

E.g. how do you rationalize \( \frac{1}{ \sqrt{2} + \sqrt[3] { 3} } \)?

Can you please explain the answer to your question?

@Rama Devi – Hello, please see the problem " Inspired by Swapnil Das".

@Swapnil Das – I couldn't understand these solutions that are posted.Can you tell an easy method? Please

@Rama Devi – Please give your response soon.

Doubt again. Why should we multiply the given expression?

he used this identity \[\displaystyle{\left( { x }^{ 3 }+1 \right) =\left( x+1 \right) \left( { x }^{ 2 }-x+1 \right) }\]

Thank You

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`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

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TopNewestLet cube root of 2 be x.

Then multiply above and below with (x^2 - x + 1).

You get the denominator as (x^3 + 1) = 3 a rational number.

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Can you explain what is the motivation behind "Multiply with \( x^2 - x + 1 \)?"

For another expression, how do we determine what to do?

E.g. how do you rationalize \( \frac{1}{ \sqrt{2} + \sqrt[3] { 3} } \)?

Log in to reply

Can you please explain the answer to your question?

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Doubt again. Why should we multiply the given expression?

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he used this identity \[\displaystyle{\left( { x }^{ 3 }+1 \right) =\left( x+1 \right) \left( { x }^{ 2 }-x+1 \right) }\]

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Thank You

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