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 a power, 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 produces 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.
Step-by-step explanation:
Hope it helps!
