Is $x^y*x^z=x^(yz)$ sometimes, always, or never true?

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Answer 1 · 2017-07-27T22:50:49

$x^y*x^z = x^(yz)$ is sometimes true.

If $x = 0$ and $y, z > 0$ then:

$x^y*x^z = 0^y*0^z = 0*0 = 0 = 0^(yz) = x^(yz)$

If $x != 0$ and $y = z = 0$ then:

$x^y*x^z = x^0*x^0 = 1*1 = 1 = x^0 = x^(0*0) = x^(yz)$

If $x = 1$ and $y, z$ are any numbers then:

$x^y*x^z = 1^y*1^z = 1*1 = 1 = 1^(yz) = x^(yz)$

It does not hold in general.

For example:

$2^3*2^3 = 2^6 != 2^9 = 2^(3*3)$

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Footnote

The normal "rule" for $x^y*x^z$ is:

$x^y*x^z = x^(y+z)$

which generally holds if $x != 0$

Is x^y*x^z=x^(yz)x^y*x^z=x^(yz) sometimes, always, or never true?