Why Is an Exponent of 1/2 the Same as a Square Root?

Date: 05/05/2004 at 09:05:30
From: Will
Subject: Algebra 2

Why is raising a number to the 1/2 power the same as taking the square
 root of the number?  For example, 36^(1/2) = 6.

It's not confusing, Iam just wondering why that is the only number
that can do that.  It seems pretty interesting to me, so could you
please answer?

Date: 05/05/2004 at 14:03:02
From: Doctor Terrel
Subject: Re: Algebra 2

Hi Will -

To show you why the "1/2 power" of a number means the same as "square
root" of the number, I need to give a little background first.  Let's
try this...

The square root of 9 is 3 because 3^2 = 3 * 3 = 9.

The square root of 64 is 8 because 8^2 = 8 * 8 = 64.

In summary, the square root of some number N is a value that when 
multiplied by itself (or squared) produces the given number N.  Or in
other words, r will be the square root of N if r^2 or r * r = N.

Are you familiar with the basic laws of exponents?  Let's apply some
of them to this question.  We'll start with:

  x^a * x^b = x^(a + b)

This law says that when you multiply like bases, you keep the base and
add the exponents, as shown here:

  x^2 * x^3 = (x*x)*(x*x*x) = x*x*x*x*x = x^5, which is x^(2 + 3)

Let's try using that law with the 1/2 power:

  x^(1/2) * x^(1/2) = x^(1/2 + 1/2) = x^1 = x

But look at what just happened!  We multiplied x^(1/2) by itself (or
squared it) and we got x.  According to our earlier summary/definition
of the square root, that means that x^(1/2) must be the square root of
x.  Can you see that?

Here's one more way to look at it using exponent laws.  There is a law
that says that a power raised to a power is the product of the powers.
In other words:

  (x^2)^3 = x^(2*3) or x^6

This is actually an extension of the exponent addition rule we already
looked at, since

  (x^2)^3 = x^2 * x^2 * x^2 = x^(2 + 2 + 2) = x^6

Let's suppose for a moment that we don't know how to write a square
root as an exponent, and we'll try to figure out what would work.  We
know that when we square the square root of x we will get x, as we
defined above.  So if we let n be this unknown exponent that
represents the square root of x, we know that 

  (x^n)^2 = x

Applying our "power to a power" rule, we can rewrite that equation as

  x^(2n) = x^1

Since the bases are the same on each side of the equation (x), and the
two quantities are equal, the exponents must also be the same:

  2n = 1
   n = 1/2

Ahah!  The mystery exponent n that represents the square root turns
out to be 1/2.

You can see why it works:

  [x^(1/2)]^2 = x^[(1/2) * 2] = x^1 = x

Going back to your example:

  [36^(1/2)] * [36^(1/2)] = [36^(1/2)]^2 = 36^[(1/2) * 2] = 36^1 = 36

Since multiplying 36^(1/2) by itself (or squaring it) gave 36,
36^(1/2) must in fact be the square root of 36.

Hope this helps.  Good luck.  Write again if need be...

- Doctor Terrel, The Math Forum
  http://mathforum.org/dr.math/