If the value of sqrt{-5-12 i}+sqrt{7+24 i} is | vedclass

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(C) Let $Z = \sqrt{-5-12 i} + \sqrt{7+24 i}$.We use the formula $\sqrt{a+ib} = \pm \left( \sqrt{\frac{|Z|+a}{2}} + i \frac{b}{|b|} \sqrt{\frac{|Z|-a}{

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$-6$

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(C) Let $Z = \sqrt{-5-12 i} + \sqrt{7+24 i}$.We use the formula $\sqrt{a+ib} = \pm \left( \sqrt{\frac{|Z|+a}{2}} + i \frac{b}{|b|} \sqrt{\frac{|Z|-a}{2}} \right)$.For $\sqrt{-5-12 i}$,$|Z| = \sqrt{(-5)^2 + (-12)^2} = 13$. Thus,$\sqrt{-5-12 i} = \pm \left( \sqrt{\frac{13-5}{2}} - i \sqrt{\frac{13+5}{2}} \right) = \pm(2-3i)$.For $\sqrt{7+24 i}$,$|Z| = \sqrt{7^2 + 24^2} = 25$. Thus,$\sqrt{7+24 i} = \pm \left( \sqrt{\frac{25+7}{2}} + i \sqrt{\frac{25-7}{2}} \right) = \pm(4+3i)$.Given $Z = \pm(2-3i) \pm(4+3i)$.Possible values for $Z$ are $(2-3i) + (4+3i) = 6$,$(2-3i) - (4+3i) = -2-6i$,$-(2-3i) + (4+3i) = 2+6i$,and $-(2-3i) - (4+3i) = -6$.Since $k$ is a negative real number,$k = -6$.

If the value of $\sqrt{-5-12 i}+\sqrt{7+24 i}$ is a negative real number $k$,then $k=$

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