Question

HELP! LONG ANSWER!! SHOW ALL WORK / INCLUDE FINAL STATEMENT!

Using the laws of exponents, show that the following expression is true.

((a³ • b⁵)⁸)¹/² • (a² • c³)⁵ • (c⁵ • b) ⁻² • a³
___________________________ = 1
(a³ • c⁻¹)⁻⁵ • (b²)⁻⁷• ((b⁴ • a⁵)²⁴) ¹/³

Answer 1

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