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 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. Hope this helps. Good luck. Write again if need be... - Doctor Terrel, The Math Forum http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.