Recall some properites of exponentiation:
[tex]a^b\times a^c=a^{b+c}[/tex]
[tex]\left(a^b\right)^c=a^{b\times c}[/tex]
So for
(-2 x ² y ⁻¹) ⁻² (5 x ³ y) ²
first distribute the outermost exponents across each product:
((-2)⁻² (x ²)⁻² (y ⁻¹)⁻²) (5² (x ³)² y ²)
Simplify the powers using the multiplication property (the second one listed above) of exponents:
((-2)⁻² x ⁻⁴ y ²) (5² x ⁶ y ²)
Combine powers of the same terms using the addition property (the first one):
(-2)⁻² 5² x ⁻⁴⁺⁶ y ²⁺²
(-2)⁻² 5² x ² y ⁴
Finally, simplify the coefficient:
(-2)⁻² = 1 / (-2)² = 1/4
5² = 25
So you end up with
25/4 x ² y ⁴
