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Question Number 128030 by Agnibhoo last updated on 03/Jan/21

$$99\times 99=9801$$$$999\times 999=998001$$$$9999\times 9999=99980001$$$$99999\times 99999=?$$$$999999\times 999999=?$$

Answered by Geovanek last updated on 03/Jan/21

$$99=100-1={10}^2-1$$$$999=1000-1={10}^3-1$$$$99...99\left(n\mathrm{nines}\right)={10}^n-1$$$$\mathrm{So},\mathrm{we}\mathrm{have}$$$${(99...99)}^2={({10}^n-1)}^2$$$$={10}^{2n}-2\cdot {10}^n+1$$$$\mathrm{For}n=5,\mathrm{we}\mathrm{have}$$$${99999}^2={10}^{10}-2\cdot {10}^5+1$$$${99999}^2=10000000000-200000+1$$$${99999}^2=9999800001$$$$\mathrm{For}n=6,\mathrm{we}\mathrm{have}$$$${999999}^2=999998000001$$$$\mathrm{You}\mathrm{put}n-1‘‘{\mathrm{nines}}^{″}\mathrm{and}n-1‘‘{\mathrm{zeros}}^{″}.$$

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