Remember: xᵃ. xᵇ = xᵃ⁺ᵇ
and (xᵃ)ᵇ = xᵃᵇ
For easy understanding, let's solve each parenthesis separately, starting with the numerator: (you start eliminating the external parenthesis
:((a³ • b⁵)⁸)¹/² • (a² • c³)⁵ • (c⁵ • b) ⁻² • a³
((a³ • b⁵)⁸)¹/² = (a³ • b⁵)⁴ = a¹² • b²⁰ (1)
(a² • c³)⁵ = a¹⁰ • c¹⁵ (2)
(c⁵ • b) ⁻² • a³ = c⁻¹⁰ • b⁻² • a³ (3)
Now (1).(2).(3) = (a¹² • b²⁰) . (a¹⁰ • c¹⁵).( c⁻¹⁰ • b⁻² • a³)=
a¹²⁺¹⁰⁺³.b²⁰⁻².c¹⁵⁻¹⁰ = a²⁵.b¹⁸.c₅ (4) . ((4) is the numerator simplified)
Let's proceed the same way with the denominator:
(a³ • c⁻¹)⁻⁵ • (b²)⁻⁷• ((b⁴ • a⁵)²⁴) ¹/³
(a³ • c⁻¹)⁻⁵ = a⁻¹⁵.c⁵ (5)
(b²)⁻⁷ = b⁻¹⁴ (6)
((b⁴ • a⁵)²⁴) ¹/³ = (b⁴ • a⁵)⁸ = b³².a⁴⁰ (7)
Now (5).(6).(7) = (a⁻¹⁵.c⁵).(b⁻¹⁴).(b³².a⁴⁰) =
a⁻¹⁵⁺⁴⁰.b⁻¹⁴⁺³².c⁵ =
a²⁵.c¹⁸.c⁵ (8) ((8) is the denominator simplified)
At last (4)/(8) = a²⁵.b¹⁸.c⁵ / a²⁵.c¹⁸.c⁵ =1 Since numerator = denominator
