$5^1=5$       

$5^0=1$  

${\left(-5\right)}^1=-5$  

$1^5=1\cdot 1\cdot 1\cdot 1\cdot 1=1$  

$0^5=0\cdot 0\cdot 0\cdot 0\cdot 0=0$  

${\left(-1\right)}^1=-1$  

${\left(-5\right)}^0=1$  

$0^1=0$  

${\left(-1\right)}^5=\left(-1\right)\cdot \left(-1\right)\cdot \left(-1\right)\cdot \left(-1\right)\cdot \left(-1\right)=1\cdot 1\cdot \left(-1\right)=-1$  

${\left(-1\right)}^0=1$  

$1^0=1$