Math Is Fun Forum

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

(Number)^1 = Number

Let a = any Number

Prove that a^1 = a

Bob

Administrator

Registered: 2010-06-20

Posts: 10,637

Re: (Number)^1 = Number

All of these power questions are answered the same way.

We start with what do we mean by a^b.

If b is a positive whole number then a^b = a x a x a x a x .... x a where a occurs b times in the multiplication.

This leads to some rules for combining powers.

eg a^3 x a^4 = (a x a x a) x (a x a x a x a) = a x a x a x a x a x a x a = a^7

So rule 1 is a^n x a^m = a ^(n+m) where n and m are positive whole numbers.

In the same way a^n ÷ a^m  = a^(n-m) if n>m

and (a^n)^m = a^(nm)

Once these rules are established for n and m being positive whole numbers mathematicians decided to declare that the rules should apply if n and m are any real numbers.  It makes sense to make the rules the defining thing about these other powers. It means you can combine numbers involving powers with one set of rules.

So now to answer your question.

a^n x a^1 = a^(n+1) by rule one.  But also a^n x a = a^(n+1). That means that a^1 behaves like a  so it is defined to be that.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: (Number)^1 = Number

All of these power questions are answered the same way.

We start with what do we mean by a^b.

If b is a positive whole number then a^b = a x a x a x a x .... x a where a occurs b times in the multiplication.

This leads to some rules for combining powers.

eg a^3 x a^4 = (a x a x a) x (a x a x a x a) = a x a x a x a x a x a x a = a^7

So rule 1 is a^n x a^m = a ^(n+m) where n and m are positive whole numbers.

In the same way a^n ÷ a^m  = a^(n-m) if n>m

and (a^n)^m = a^(nm)

Once these rules are established for n and m being positive whole numbers mathematicians decided to declare that the rules should apply if n and m are any real numbers.  It makes sense to make the rules the defining thing about these other powers. It means you can combine numbers involving powers with one set of rules.

So now to answer your question.

a^n x a^1 = a^(n+1) by rule one.  But also a^n x a = a^(n+1). That means that a^1 behaves like a  so it is defined to be that.

Bob

Very informative. Very detailed.  I thank you for your time and effort.