My Math Forum Definition of Square root and Exponents.

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 June 18th, 2009, 12:59 AM #1 Newbie   Joined: Feb 2009 Posts: 3 Thanks: 0 Definition of Square root and Exponents. Hi all! I am a Math newb... The thing is I've gotten stuck with the thought of that any number with the exponent 0.5 equals the numbers square root right? And the regular definition of the exponent is that 5^5 equals 5*5*5*5*5.. right? But then I am wondering about the real definition of exponent.. since you can't really write 5*5 one half time right? Or am I wrong? (I'm thinking of 5^0.5) So my thought is that 5^1 = 5*5 one whole time, and 5^0.5 = 5*5 one half time????? Well.. I've been thinking of this for some time now.. And by myself I can't really come to any good answers.. And I don't really feel like studying this in books and stuff.. Since I'm not really a math expert, and wouldn't know what to look for.. or what to make of the text in a advanced math book such as that.. One more thing I'm after is how to solve a square root with pen and paper... I'm aware of the Babylonian, but I'm after an other solution to that, with the "real" definition of exponents... hmm... If you know what I mean! Cheers, Artheus
June 18th, 2009, 01:00 PM   #2
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Re: Definition of Square root and Exponents.

Well, if you can't understand it, try to prove it!
So if you have lets say you have an equation:
$a^p=a^q$
It should be obvious that $p=q$ (because the base is the same [it's a property of logarithms]). Now, a long time ago, some mathematician decided that there should be some exponential way to express roots. So he wrote down an equation to help solve it:
$\sqrt{a}=a^x$, where $x$ is the the exponential representation of square roots. We are to solve for $x$.
Let's start by squaring both sides:
$\left(\sqrt{a}\right)^2=\left(a^x\right)^2\\a=a^{2 x}\\a^1=a^{2x}$
Now, using the property we discovered earlier, we get:
$1=2x\\x=\frac{1}{2}$
And so,
$\sqrt{a}=a^{\frac{1}{2}}$
You can prove this for any root $r$ and power $p$. You would always get $\frac{p}{r}$.

Quote:
 Originally Posted by artheus One more thing I'm after is how to solve a square root with pen and paper
There's multiple ways to do this. When you get to Calculus you will figure out a way to approximate square roots, and that is my preference.
To start, pick a perfect square that is close to your number. For example, if the number you want to square root is 150, 144 (the square of 12) is a perfect square close to 150. Call that perfect square $x_1$. Now plug it all it into the equation:
$\sqrt{x}\approx \frac{x+x_1}{2\sqrt{x_1}}$
This formula returns 12.25 for the approximation of $\sqrt{150}$, which is just about 0.00255128 off from the real value.
Also, take a look at this method.

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