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

Jul 27, 2017

${x}^{y} \cdot {x}^{z} = {x}^{y z}$ is sometimes true.

#### Explanation:

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

${x}^{y} \cdot {x}^{z} = {0}^{y} \cdot {0}^{z} = 0 \cdot 0 = 0 = {0}^{y z} = {x}^{y z}$

If $x \ne 0$ and $y = z = 0$ then:

${x}^{y} \cdot {x}^{z} = {x}^{0} \cdot {x}^{0} = 1 \cdot 1 = 1 = {x}^{0} = {x}^{0 \cdot 0} = {x}^{y z}$

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

${x}^{y} \cdot {x}^{z} = {1}^{y} \cdot {1}^{z} = 1 \cdot 1 = 1 = {1}^{y z} = {x}^{y z}$

It does not hold in general.

For example:

${2}^{3} \cdot {2}^{3} = {2}^{6} \ne {2}^{9} = {2}^{3 \cdot 3}$

$\textcolor{w h i t e}{}$
Footnote

The normal "rule" for ${x}^{y} \cdot {x}^{z}$ is:

${x}^{y} \cdot {x}^{z} = {x}^{y + z}$

which generally holds if $x \ne 0$