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Question Number 202044 by hardmath last updated on 19/Dec/23

$$\mathrm{If}$$$$\mathrm{a}=9999999998000000001$$$$\mathrm{Find}$$$$\mathrm{A}=\sqrt[9]{\sqrt{\mathrm{a}}+1}\to \mathrm{A}=?$$

Commented by mr W last updated on 19/Dec/23

$$doyouhaveacalculator?{you}^{\prime }llget$$$$A\approx 11.364$$$$pleasecheckyourquestionagain,$$$$maybeyouhavewrittenone9too$$$$much!$$

Answered by esmaeil last updated on 19/Dec/23

$$\sqrt[9]{\sqrt{99999.99999\ast 99999.99999\ast {10}^8}+1}=$$$$\sqrt[9]{99999.99999\ast {10}^4+1}=$$$$\sqrt[9]{999999999.9+1}\cong 10$$

Answered by mr W last updated on 20/Dec/23

$${99}^2=9801$$$${999}^2=998001$$$$...$$$$\underset{ntimes}{999...9^2}={({10}^n-1)}^2=({10}^n-2)\times {10}^n+1=\underset{n-1}{999...98}\underset{n-1}{000...01}$$$$$$$$a=\underset{8times}{999999998}\underset{8times}{000000001}=999999{999}^2$$$$\sqrt{a}=999999999$$$$\sqrt{a}+1=999999999+1=1000000000={10}^9$$$$A=\sqrt[9]{\sqrt{a}+1}=\sqrt[9]{{10}^9}=10✓$$

Commented by hardmath last updated on 20/Dec/23

$$\mathrm{cool}\mathrm{dear}\mathrm{professor}\mathrm{thankyou}$$

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