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How can $4^x = 4^{400} + 4^{400} + 4^{400} + 4^{400}$ have the solution $x = 401$?

Can someone explain to me how this works in a simple way?

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    $\begingroup$ What a stark contrast when people are voting to close this question but a similar one has received over a hundred upvotes ... $\endgroup$ – user1551 May 30 '13 at 23:51
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    $\begingroup$ @user1551, don't bother rationalizing upvote patterns on this site; you'll just go nuts. $\endgroup$ – J. M. is a poor mathematician Jun 1 '13 at 6:00
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    $\begingroup$ @user1551: It seems you need to preface every "basic" question with a reference to your 5 yo niece or a disclaimer regarding your level of education. $\endgroup$ – Najib Idrissi Jun 25 '13 at 8:45
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$4^{401} = 4\times 4^{400} = 4^{400} + 4^{400} + 4^{400} + 4^{400}$

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First remember that $4^x$ means you multiply $4$ together with itself $x$ times. For example $4^2 = 4*4= 16.$ Note that $4^{400} + 4^{400}+4^{400}+4^{400} = 4^{400}(1+1+1+1) = 4^{400}*4$. And $4^{400}*4$ is just $4$ multiplied together with itself 401 times, which is the same as writing $4^{401}$.

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$$4^{401}=4^{400+1}=4^{400}\cdot 4^1=4\cdot 4^{400}$$

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It only works because $x=401$.

$4^{401}$ means multiply $4$ by itself 401 times. So $$\begin{array}{rl} 4^{401} &= 4\times 4^{400} \\ &= 4^{400} +4^{400} +4^{400} +4^{400} \end{array}$$

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$a=4^{400}$

$4^x=a+a+a+a$ $$4^x=4\cdot a$$ $$4^x=4\cdot 4^{400}$$ $$u^m\times u^n= u^{m+n}\implies 4^1\times 4^{400}=4^{401}$$ $$4^x=4^{400+1}$$ $$a^m=a^n \implies m=n$$ $$x=401$$

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The number you're adding is being added n (number) times. Well, we can infer from this that if that number is being added n (number) times, it is multiplicating itself. Now, if a number is multiplicating itself, then we have an exponentiation!

$N_1+N_2+N_3+...+N_N = N\times N = N^2$

If you're adding a number, no matter how big it is or what operation you're doing with it, and you're adding it n (number) times, you'll end up with

$(N^k)1+(N^k)2+(N^k)3+...(N^k)N = (N^k)N$

Which is $N^{k+1}$

So, if you're doing it with 4^400, we've got

$(4^{400})+(4^{400})+(4^{400})+(4^{400}) = (4^{400})\times 4 = 4^{401}$

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