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Question Number 60620 by naka3546 last updated on 22/May/19

$$4^x+9^x+{25}^x=6^x+{10}^x+{15}^x$$$$x=?$$

Answered by fjdjdcjv last updated on 22/May/19

$$x=0$$

Answered by tanmay last updated on 22/May/19

$$2^x=a{3}^x=b{5}^x=c$$$$a^2+b^2+c^2=ab+bc+ca$$$$2a^2+2b^2+2c^2-2ab-2bc-2ac=0$$$${(a-b)}^2+{(b-c)}^2+{(c-a)}^2=0$$$$so{(a-b)}^2=0\to a=b$$$$similarly{(b-c)}^2=0\to b=c$$$$soa=b=c$$$$now{2}^x=3^x$$$${\left(\frac{2}{3}\right)}^x=1={\left(\frac{2}{3}\right)}^0\mathit{s}\mathit{o}\mathit{x}=0$$

Answered by mr W last updated on 24/May/19

$$a^x+b^x+c^x=p^x+q^x+r^x$$$$or$$$$a^x+b^x+c^x+d^x=p^x+q^x+r^x+s^x$$$$......$$$$\Rightarrow x=0is\mathit{o}\mathit{n}\mathit{e}solution$$

Commented by naka3546 last updated on 23/May/19

$$Thankyou,sir.$$

Commented by fjdjdcjv last updated on 23/May/19

$$thereare3solutions$$$$butithinkthese$$$$1+5^x+8^x+{12}^x=2^x+3^x+{10}^x+{11}^x$$$$withx=1x=2andx=3$$

Commented by mr W last updated on 24/May/19

$$youarerightsir,thanks!$$

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