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# Understanding square roots

Learn how square root means what number multiplied by itself will result in the given number. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• would you please explain why we have to use this square roots?
and please explain an example that where this square roots can be used? •  You will also learn Pythagoras' Theorem, which helps you find the length of one side of some triangles if you know the other sides. Carpenters or builders use this idea to make sure walls and floors are straight. Squares and square roots are also a part of the real world in the way circles and other curvy shapes are described mathematically.

You can use it to find out if you're getting the best price for pizza, too. The area of a circle is pi x the radius squared.
• I understand what a square root is but is there an easier way to find it for any number? I usually end up googling it because it is a lot of work •   There is in fact a way to find square roots without a calculator. It's the so called "guess and check" method where you basically estimate. If you are asked to find the square root of 30, for example, you know that 5 squared is 25 and 6 squared is 36, so the final result will be somewhere in between that. You could just say 5.5. That's actually very close. If you need more accuracy, you'll continue guessing slightly lower and slightly higher numbers to narrow down your results. In case you are wondering what the square root of 30 is, though, I will say that it is about 5.48 or so, rounded to two decimal digits.

You definitely should memorize the common perfect squares. A perfect square is a number that has a square root that is a whole number. 30 is not a perfect square because its square root IS NOT a whole number, but 36 is because its square root is 6, which is a whole number. I'll list the first thirteen or fourteen perfect squares.

1. Square root: 1
4. Square root: 2
9. Square root: 3
16. Square root: 4
25. Square root: 5
36. Square root: 6
49. Square root: 7
64. Square root: 8
81. Square root: 9
100. Square root: 10
121. Square root: 11
144. Square root: 12
169. Square root: 13
196. Square root: 14
225. Square root: 15

You can find higher perfect squares if you wish simply by entering some whole number into the calculator, then multiplying it by itself. If, for example, you wanted to find the square of 16, you would multiply it by 16 in the calculator and get 256, which is a perfect square, and so on... This works with higher numbers like 65536 as well, where the square root so happens to be 256 (although you don't need to memorize numbers that are quite so high LOL). It is a very, very, very good idea to have the most common perfect squares memorized. Probably you don't need to find the perfect squares above the ones I listed unless you want to. :)

Hope this helps. Also, giving credit where credit is due, I actually looked this up to answer your problem. I would strongly recommend reading this: http://www.homeschoolmath.net/teaching/square-root-algorithm.php
• Is it like finding 10% of a number? I'm kind of confused with that a little... • what does one do if they are adding two square roots, such as the square root of 13 + the Square root of 13? • Is there is a trick or shortcut into finding the answer of a square root, other than guess and check? • Most square roots are irrational, meaning that their decimal form continues forever without a repeating pattern. If you are trying to take the square root of a number that is not a perfect square, the best you can hope for is an approximation. You are usually best served to use a calculator to get these results, but there is a method I enjoy for approximating square roots. It is an iterative method developed by Heron of Alexandria, an ancient Greek engineer.

First, guess a convenient value for the square root. Divide the number by your guess. Now you have two numbers that multiply to get your original number. Take the average of these two numbers. This becomes your second guess for the square root. So again, you can divide the original number by this new guess, and take the average of these two numbers to get a third guess, and so on. Soon consecutive guesses will not change much. This is the approximation of the square root.
• So the square root of a number is the number that when you multiply it by itself is equal to that number? • So, is this how every "square root" is? Im a little confused... • Yeah think about a square root as the number you get when you multiply something by itself.

Helps to think about the definition of multiplication as adding a number to itself:

``2 x 3 = 2+2+2 = 3+3 = 6``

Exponents are similar, except now we're multiplying the number to itself instead of adding it.

``2^2 (squared) = 2 x 2 = 2+2 = 43^2 (squared) = 3 x 3 = 3+3+3 = 9``

Taking the square root is figuring out what number multiplied by itself is equal to the number under the square root symbol.

So:

``√4 = 2, because 2*2 OR 2^2 = 4√9 = 3, because 3 x 3 = 9 OR 3^2 = 9``

Hopefully that helps!
• What will we use square roots for? • I've been experimenting with square roots (sometimes on the calculator for super hard ones to solve) but I was thinking "how about negative numbers? Do they have square roots? And if there is, what is it?" For example, what is the square root of... -1? Please answer! • Good question! The answer is that there are no REAL solutions. For algebra, this means "no there are no answers." There are in fact "imaginary" numbers that result from taking the square root of a negative number. I don't know why they are called "imaginary" because they really do exist- just not in a way that is as easy to understand or write on paper. I still have trouble understanding them myself. Khan Academy has an entire series explaining imaginary numbers. It is discussed briefly right before learning calculus, and is discussed in depth at advanced math levels way beyond calculus (I still haven't taken those yet). 