Solve the following

Given:

$5^{2 x+1} \div 25=125$

We have:

$25=5 \times 5=5^{2}$

$125=5 \times 5 \times 5=5^{3}$

$\therefore \frac{5^{2 x+1}}{5^{2}}=5^{3} \Rightarrow 5^{[(2 x+1)-2]}=5^{3}$

$\therefore \frac{5^{2 x+1}}{5^{2}}=5^{3} \Rightarrow 5^{[(2 x+1)-2]}=5^{3}$

or $5^{[(2 x+1)-2]}=5^{[2 x-1]}=5^{3}$

$\Rightarrow 2 x-1=3$

$2 x=3+1=4$

$x=\frac{4}{2}=2$

$\therefore x=2$