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Category: Algebra

Question-222754

Question Number 222754 by hardmath last updated on 06/Jul/25 Commented by hardmath last updated on 06/Jul/25 $$\int_{\mathrm{0}} ^{\:\mathrm{8}} \left(\mathrm{f}^{\:−\mathrm{1}} \left(\mathrm{x}\right)\:−\:\mathrm{f}\left(\mathrm{x}\right)\right)\:\mathrm{dx}\:=\:\mathrm{12} \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{8}} \:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:? \\…

Question-222585

Question Number 222585 by Rojarani last updated on 30/Jun/25 Commented by AntonCWX8 last updated on 01/Jul/25 $${Now}\:{I}'{ll}\:{give}\:{you}\:{guys}\:{another}\:{challenge}. \\ $$$${What}\:{if}\:{the}\:{numbers}\:{beside}\:{the}\:\sqrt{\:\:}\:{symbol}\:{represent}\:{the}\:{nth}\:{root}? \\ $$$${Say}\:\mathrm{5}\sqrt{\:\:\:}\:=\:\sqrt[{\mathrm{5}}]{\:\:\:\:} \\ $$ Commented by…

If-a-i-gt-0-b-i-gt-0-i-1-n-Prove-that-a-1-2-b-1-2-a-2-2-b-2-2-a-n-2-b-n-2-a-1-a-2-a-n-2-b-1-b-2-b-

Question Number 222582 by hardmath last updated on 30/Jun/25 $$\mathrm{If}:\:\:\:\mathrm{a}_{\boldsymbol{\mathrm{i}}} \:>\:\mathrm{0}\:\:\:,\:\:\:\mathrm{b}_{\boldsymbol{\mathrm{i}}} \:>\:\mathrm{0}\:\:\:,\:\:\:\mathrm{i}\:=\:\overline {\mathrm{1},…,\mathrm{n}} \\ $$$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\sqrt{\mathrm{a}_{\mathrm{1}} ^{\mathrm{2}} \:+\:\mathrm{b}_{\mathrm{1}} ^{\mathrm{2}} }\:+\:\sqrt{\mathrm{a}_{\mathrm{2}} ^{\mathrm{2}} \:+\:\mathrm{b}_{\mathrm{2}} ^{\mathrm{2}} }\:+…+\:\sqrt{\mathrm{a}_{\boldsymbol{\mathrm{n}}}…

solve-for-p-q-s-in-terms-of-c-qs-q-sp-2-s-qs-q-sp-p-0-q-c-p-1-2-sp-q-q-cp-p-1-2-q-c-3-I-have-to-find-non-zero-real-x-q-c-p-1-

Question Number 222583 by ajfour last updated on 30/Jun/25 $${solve}\:{for}\:{p},{q},{s}\:{in}\:{terms}\:{of}\:{c}. \\ $$$$\bullet\:\left(\frac{{qs}}{{q}−{sp}}\right)^{\mathrm{2}} −{s}\left(\frac{{qs}}{{q}−{sp}}\right)+{p}=\mathrm{0} \\ $$$$\bullet\:\left(\frac{{q}+{c}}{{p}+\mathrm{1}}\right)^{\mathrm{2}} ={sp}−{q} \\ $$$$\bullet\:\left({q}−{cp}\right)\left({p}+\mathrm{1}\right)^{\mathrm{2}} =\left({q}+{c}\right)^{\mathrm{3}} \\ $$$${I}\:{have}\:{to}\:{find}\:{non}\:{zero}\:{real}\:{x}=−\left(\frac{{q}+{c}}{{p}+\mathrm{1}}\right)\:. \\ $$ Terms of…

x-x-64-x-

Question Number 222552 by mathlove last updated on 29/Jun/25 $${x}^{{x}} =\mathrm{64}\:\:\:\:\:\:,\:\:{x}=? \\ $$ Answered by mr W last updated on 29/Jun/25 $${x}\mathrm{ln}\:{x}=\mathrm{ln}\:\mathrm{64} \\ $$$$\mathrm{ln}\:{xe}^{\mathrm{ln}\:{x}} =\mathrm{6ln}\:\mathrm{2}…