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  4. The value of (52+6 √43)3 / 2-(52-6 √43)3 / 2 is

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Q. The value of $(52+6 \sqrt{43})^{3 / 2}-(52-6 \sqrt{43})^{3 / 2}$ is

Complex Numbers and Quadratic Equations

Solution:

$\left[(\sqrt{43}+3)^2\right]^{3 / 2}-\left[(\sqrt{43}-3)^2\right]^{3 / 2}$
$(3+\sqrt{43})^3+(3-\sqrt{43})^3=2\left[{ }^3 C _0 \cdot 3^3+{ }^3 C _2 \cdot 3(\sqrt{43})^2\right] $
$2[27+9 \cdot 43]=2[27+378]=2(414)=828$