Help: Algebra

$\large \dfrac1{2+\sqrt3} = 2 -\sqrt3$

Why is the equation above true?

#Algebra

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

$\dfrac{1}{n+\sqrt{n+1}}=\dfrac{n-\sqrt{n+1}}{(n+\sqrt{n+1})(n-\sqrt{n+1})}=\dfrac{n-\sqrt{n+1}}{n^2-n-1}$

We want denominator to be 1 so,

$\implies n^2-n-1=1\\ \implies n^2-n-2=0 \\ \implies n^2-2n+n-2=0 \\ \implies n(n-2)+1(n-2)=0 \\ \implies (n+1)(n-2)=0$

So it only works if $n=-1,2$ :)

Nihar Mahajan - 5 years, 1 month ago

You are smart, thx man

Jason Chrysoprase - 5 years, 1 month ago

Write $2$ as $\sqrt{4}$ ,we'll get $\frac{1}{2+\sqrt{3}}=\frac{1}{\sqrt{3}+\sqrt{4}}=\frac{4-3}{\sqrt{4}+\sqrt{3}}=\frac{(\sqrt{4})^2-(\sqrt{3})^2}{\sqrt{4}+\sqrt{3}}=\frac{(\sqrt{4}+\sqrt{3})(\sqrt{4}-\sqrt{3})}{\sqrt{4}+\sqrt{3}}=\sqrt{4}-\sqrt{3}=\boxed{2-\sqrt{3}}$ You could generalized to $\frac{1}{\sqrt{n}+\sqrt{n+1}}$ though

Pham Khanh - 5 years ago

Nice ;)

Jason Chrysoprase - 5 years ago

Rationalise the denominator

Rushabh Zambad - 5 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$