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Question Number 47993 by F_Nongue last updated on 18/Nov/18

$$Solvein\mathbb{R}$$$$\sqrt[3]{59+\sqrt{x-2}}+\sqrt[3]{12-\sqrt{x-11}}=6$$

Commented by tanmay.chaudhury50@gmail.com last updated on 18/Nov/18

$$x=27$$

Commented by ajfour last updated on 18/Nov/18

$$stepsplease..$$

Commented by MJS last updated on 18/Nov/18

$$\mathrm{we}\mathrm{can}\mathrm{try}$$$$\mathrm{or}$$$$\mathrm{we}\mathrm{can}\mathrm{eliminate}\mathrm{all}\mathrm{roots}\mathrm{which}\mathrm{leads}\mathrm{to}$$$$\mathrm{a}\mathrm{polynome}\mathrm{of}{5}^{\mathrm{th}}\mathrm{degree}\Rightarrow \mathrm{we}\mathrm{can}\mathrm{only}\mathrm{try}$$$$\mathrm{again}.\mathrm{is}\mathrm{there}\mathrm{an}\mathrm{easier}\mathrm{way}?$$$$[\mathrm{putting}x=t+\frac{13}{2}\mathrm{helps}\mathrm{while}\mathrm{transforming}]$$

Commented by MJS last updated on 18/Nov/18

$$\mathrm{trying}\mathrm{to}\mathrm{find}x\mathrm{with}\mathrm{both}x-2\mathrm{and}x-11$$$$\mathrm{are}\mathrm{square}\mathrm{numbers}\mathrm{we}\mathrm{get}x=11\mathrm{or}x=27$$$$\mathrm{but}\mathrm{only}x=27\mathrm{fits}$$$$f\left(x\right)=\sqrt[3]{59+\sqrt{x-2}}+\sqrt[3]{12-\sqrt{x-11}}-6\mathrm{is}\mathrm{decreasing}$$$$\mathrm{for}x⩾11\Rightarrow {\mathrm{there}}^{\prime }\mathrm{s}\mathrm{only}1\mathrm{solution}$$

Commented by tanmay.chaudhury50@gmail.com last updated on 18/Nov/18

$$fromproblem...$$$$1)x>2andx>11$$$$ihavesolvedlogically$$$$6=1+5$$$$6=2+4$$$$6=3+3$$$$thatmeans$$$$59+\sqrt{x-2}=64$$$$x-2=25x=27$$$$12-\sqrt{x-11}=8$$$$\sqrt{x-11}=4$$$$x=27$$

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