Power of 1?

Question

Why is any number raised to the first power = the same number?

Sample: x^1 = x, right? Now, why is that?

Answer 1

Because of the rules of ADDING exponents when you MULTIPLY numbers together, and SUBTRACTING exponents when you DIVIDE numbers into one another.

This is the advantage of logs you can add rather than multiply (easier) and subtract rather than divide.

Take the following:

(2 X 2) X (2 X 2 X 2)

If you were to do this longhand, it would be obvious that it was the same as.

2 X 2 X 2 X 2 X 2

Now let's put it into log form ( to the base 2).

In the first expression, you have 2^2 X 2^3

Is the answer 2^6? NO! it's 2^5 i.e. you ADD the exponents.

Similarly with dividing:

If you divide 2^3 by 2^3 you get 1, obviously. Using the subtraction rule, you get 2^(3-3) = 2^0 which is NOT zero, but 1.

Similarly, if you divide 2^3 by 2^2, you get 2. Using the exponent rule, you get 2^(3 - 2) = 2^1 which MUST be 2.

I hope this helps you see the logic.

Answer 2

Because x^2 = x*x

And for any natural number, n: x^(n + 1)=x * (x^n)

And thus x^(m + n) = x^m * x^n
x*x = x^2 = x * (x^1)

If x=0:

0 = x^2 = (x^1) * (x^1)
Thus x^1 = 0 = x

Otherwise,:
x^2 = x*(x^1) // divide by x

x = x^1

Answer 3

x to the power means that you're multiplying x that many times.

X^3 is x times x times x

X^1 is just x

Answer 4

x^0 means 1 for any x<>0. x^2=x*x ... that means x multiplied with x by 2 times. and.. if you multiply x just once.. x can't change.. so.. x^1=x :)

Answer 5

x^1 = x, because
x^1= [x^(n+1)] / [x^(n)] = x[x^(n)] / [x^(n)] = x
example
2^1 = 2^6 / 2^5 =(2x2x2x2x2x2) / (2x2x2x2x2)=2
and 2^(-1) = 1/2

Answer 6

1 multiplied by x means you are adding 1 x-times.

x^1 means you're multiplying x once.

Answer 7

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