Tarski's high school algebra problem

Consider the following eleven axioms about addition, multiplication, and exponentiation:

x+y𝑥𝑦\displaystyle x+y =y+xabsent𝑦𝑥\displaystyle=y+x
(x+y)+z𝑥𝑦𝑧\displaystyle(x+y)+z =x+(y+z)absent𝑥𝑦𝑧\displaystyle=x+(y+z)
x⋅1⋅𝑥1\displaystyle x\cdot 1 =xabsent𝑥\displaystyle=x
x⋅y⋅𝑥𝑦\displaystyle x\cdot y =y⋅xabsent⋅𝑦𝑥\displaystyle=y\cdot x
(x⋅y)⋅z⋅⋅𝑥𝑦𝑧\displaystyle(x\cdot y)\cdot z =x⋅(y⋅z)absent⋅𝑥⋅𝑦𝑧\displaystyle=x\cdot(y\cdot z)
x⋅(y+z)⋅𝑥𝑦𝑧\displaystyle x\cdot(y+z) =x⋅y+x⋅zabsent⋅𝑥𝑦⋅𝑥𝑧\displaystyle=x\cdot y+x\cdot z
1xsuperscript1𝑥\displaystyle 1^{x} =1absent1\displaystyle=1
x1superscript𝑥1\displaystyle x^{1} =xabsent𝑥\displaystyle=x
xy+zsuperscript𝑥𝑦𝑧\displaystyle x^{y+z} =xy⋅xzabsent⋅superscript𝑥𝑦superscript𝑥𝑧\displaystyle=x^{y}\cdot x^{z}
(x⋅y)zsuperscript⋅𝑥𝑦𝑧\displaystyle(x\cdot y)^{z} =xz⋅yzabsent⋅superscript𝑥𝑧superscript𝑦𝑧\displaystyle=x^{z}\cdot y^{z}
(xy)zsuperscriptsuperscript𝑥𝑦𝑧\displaystyle(x^{y})^{z} =xy⋅zabsentsuperscript𝑥⋅𝑦𝑧\displaystyle=x^{y\cdot z}

Tarski's high school algebra problem asks if these set of axioms are Complete for the theory of non-zero Natural Numbers; EG: are there any identities involving only addition, multiplication, and exponentiation that are true for non-zero Natural Numbers, yet not provable from the axioms above. Wilkie's Identity is an example of a non-provable identity, though there are an infinite number of examples: the Gurevic Identities.