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Question Number 213002 by efronzo1 last updated on 28/Oct/24

$$\frac{{(52+6\sqrt{43})}^{3/2}-{(52-6\sqrt{43})}^{3/2}}{18}=?$$

Answered by Rasheed.Sindhi last updated on 28/Oct/24

$$\frac{{(52+6\sqrt{43})}^{3/2}-{(52-6\sqrt{43})}^{3/2}}{18}=?$$$$\sqrt{52+6\sqrt{43}}=\sqrt{43}+3$$$$\sqrt{52-6\sqrt{43}}=\sqrt{43}-3$$$$\frac{{(\sqrt{43}+3)}^3-{(\sqrt{43}-3)}^3}{18}$$$$a^3-b^3=(a-b)(a^2+ab+b^2)$$$$\frac{(\sqrt{43}+3-\sqrt{43}+3)({(\sqrt{43}+3)}^2+(\sqrt{43}+3)(\sqrt{43}-3)+{(\sqrt{43}-3)}^2)}{18}$$$$=\frac{6(52+6\sqrt{43}+43-9+52-6\sqrt{43})}{18}$$$$\frac{6(52+34+52)}{18}=\frac{6\times 138}{18}=\frac{138}{3}=46$$

Commented by golsendro last updated on 28/Oct/24

$$\sqrt{52-6\sqrt{43}}=\sqrt{43}-3$$

Commented by Rasheed.Sindhi last updated on 28/Oct/24

$$\mathcal{T}hankssir,I^{\prime }vecorrected.$$

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