Power of a Power Property

To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, (a^m)^n will be rewritten as a product using the definition of a power. (a^m)^n=a^m * a^m * ⋯ * a^m_(ntimes) Then, all occurrences of the expression a^m will be written as products using the definition of a power one more time. (a^m)^n=a* ⋯ * a_(m times) * a* ⋯ * a_(m times) * ⋯ * a* ⋯ * a_(m times) ↘ ↓ ↙ n times Notice that the base a is multiplied by itself m* n times. By the definition of a power, the right-hand side of the above equation is equivalent to raising a to the power of m* n. a * a * a * ⋯ * a_(m* ntimes) =a^(m* n) The Power of a Power Property has been proved.