Is there a way to simplify $x ^ y + x ^ z$ to a formula with only one $x$?
I know $(x ^ y)(x ^ z) = (x ^ {y + z})$, but how can it change in addition?
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2 Answers
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You can't rewrite $x^y+x^z$ to something with only one $x$.
Assume that $x^y+x^z=x^{f(y,z)}$.
Then $$ 24=8+16=2^3+2^4=2^{f(3,4)} $$ giving us $f(3,4)=\log_2(24)=4,584\ldots$
and $$ 108=27+81=3^3+3^4=3^{f(3,4)} $$ giving us $f(3,4)=\log_3(108)=4,2618\ldots$
That's a contradction.
answered Dec 22, 2014 at 23:55
Henrik supports the community
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Addition doesn't behave nicely with exponentiation, like your multiplication example shows. If you want a single operation, you can always define $\text{expadd}_{y,z}(x):=x^y+x^z$, then you have a single formula: $$\text{expadd}_{y,z}(x)$$ Somehow, I doubt that's what you wanted.
