The following is a recursive definition of a^n: a^1 =a...

The following is a recursive definition of $a^{n}:$ $$\begin{aligned}a^{1} &=a \\a^{n+1} &=a^{n} \cdot a \end{aligned}$$ Prove, by induction, that$$a^{n+m}=a^{n} \cdot a^{m}$$ $$\left(a^{n}\right)^{m}=a^{n m}$$. (Don't try to be fancy: use either induction on $n$ or induction on $m,$ not both at once.)

The following is a recursive definition of $a^{n}:$ $$\begin{aligned}a^{1} &=a \\a^{n+1} &=a^{n} \cdot a
\end{aligned}$$ Prove, by induction, that$$a^{n+m}=a^{n} \cdot a^{m}$$ $$\left(a^{n}\right)^{m}=a^{n m}$$.
(Don't try to be fancy: use either induction on $n$ or induction on $m,$ not both at once.)

Key Concepts

Properties of Exponents

The properties of exponents include rules such as the product-of-powers rule and the power-of-a-power rule, which state that a^(n+m) = a^n * a^m and (a^n)^m = a^(nm) respectively. These properties are derived from the definition of exponentiation, often using either repeated multiplication or induction proofs, and they play an essential role in simplifying expressions and solving equations involving exponential functions.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, called the base, to the power of an exponent. It represents repeated multiplication of the base. For example, expressing a power a^n corresponds to multiplying a by itself n times. This operation is fundamental in various areas of mathematics and its properties, such as laws of exponents, are crucial for higher-level algebra.

Mathematical Induction

Mathematical induction is a proof technique used to establish that a proposition holds for all natural numbers. It involves proving a base case (usually for the number 1) and then showing that if the statement holds for an arbitrary positive integer n, it must also hold for n+1. This method is widely employed in proofs related to sequences, series, and properties of operations defined on the natural numbers.

Recursive Definition

A recursive definition specifies an object in terms of itself, using one or more previously defined values. In the context of exponentiation, the recursive definition establishes the power a^n by setting an initial condition (such as a^1 = a) and defining subsequent powers based on multiplication by a. This approach helps formalize the concept of exponentiation from a base case.

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