operations with powers

-  Power of a product: (a · b)ⁿ = aⁿ · bⁿ

Example:  2⁵ · 5⁵ = (2 · 5)⁵ = 10⁵ = 100.000

-  Power of a quotient: (a : b)ⁿ = aⁿ : bⁿ

Example: 12⁴ : 3⁴ = (12 : 3)⁴ = 4⁴ = 256

- Multiplication of powers with the same base: aⁿ · a^m = a^m+n        

Example:  
3² · 3³ = (3 · 3) . (3 · 3 · 3) = 3²⁺³ = 3⁵ = 243

- Division of powers with the same base: aⁿ : a^m = a^m-n       

Example: 
12²⁵ : 12²³ = 12^25-23 = 12² = 144

-  Power of a power: (a^m)ⁿ = a^m·n

Example:
(2³)² = 2³ · 2³ = (2 · 2 · 2)·(2 · 2 · 2) = 2^3·2 = 2⁶ = 64

-  NOTE:  a⁰ = 1        a¹ = a

Exercise:Reduce to one power and calculate if it is possible:                      

Solutions: a) 2^₅ = 32; b) 2⁹ = 512; c) 10⁴ = 10.000;  d) 12¹ = 12; e) x⁷; f) x⁰ = 1 g) 3⁴ = 81

h) a¹⁵; i) k²⁸; j) a¹ = a; k) x⁸; l) a⁰ = 1; m) m¹ = m; n) x³