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  4. The value of |√4+2√3|-|√4-2√3| is

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Q. The value of $\left|\sqrt{4+2\sqrt{3}}\right|-\left|\sqrt{4-2\sqrt{3}}\right|$ is

KEAMKEAM 2017Complex Numbers and Quadratic Equations

Solution:

We have,
$|\sqrt{4+2 \sqrt{3}}|-|\sqrt{4-2 \sqrt{3}}|$
$=|\sqrt{3+1+2 \sqrt{3}}|-|\sqrt{3+1-2 \sqrt{3}}|$
$=\left|\sqrt{(\sqrt{3})^{2}+(1)^{2}+2 \cdot \sqrt{3} \cdot 1}\right|$
$-\mid \sqrt{(\sqrt{3})^{2}+(1)^{2}-2 \cdot \sqrt{3} \cdot 1 \mid}$
$=\left|\sqrt{(\sqrt{3}+1)^{2}}\right|-\mid \sqrt{(\sqrt{3})-1)^{2} \mid}$
$=|\sqrt{3}+1|-|\sqrt{3}-1|$
$=(\sqrt{3}+1)-(\sqrt{3}-1)=2$