While the intimidating square root symbol can make many students nauseous, square root operations are not as difficult to solve as they may seem at first glance. Operations with simple square roots can often be solved just as easily as basic multiplications and divisions. More complex square roots, on the other hand, can take a little more work, but with the right method they too can become easy to extract. Start practicing square roots today to learn this radical new math skill!
Steps
Part 1 of 3: Understanding Squares and Square Roots
Step 1. The square of a number is the result of multiplying it by itself
To understand square roots, it's usually best to start with squares. Squares are simple to understand: squaring a number just means multiplying it by itself. For example, 3 squared is the same as 3 × 3 = 9, while 9 squared is equal to 9 × 9 = 81. Squares are written with a small "2" at the top right of the multiplied number, like this: 32, 92, 1002, and so on.
Try squaring a few more numbers on your own to see if you have the best understanding of the concept. Remember, squaring a number simply means multiplying it by itself. You can also do it with negative numbers, the result will always be positive. For example: -82 = -8 × -8 = 64.
Step 2. For square roots, find the "inverse" of a square
The square root symbol (√, also called "radical") basically represents the "opposite" operation to that of the symbol 2. When you see a radical, you will have to ask yourself, "What number can be multiplied by itself to give the number under the root as a result?" For example, if you see √ (9), you will need to find the number that can be squared to get 9. In this case, the answer is three, because 32 = 9.
-
As a further example, let's try to find the square root of 25 (√ (25)), that is the number that squared gives 25. Since 52 = 5 × 5 = 25, we can say that √ (25) =
Step 5..
-
You can also think of this process as "undoing" a square. For example, if you want to find √ (64), the square root of 64, start thinking of 64 as 82. Since the symbol of a square root, in essence, "eliminates" that of a square, we can say that √ (64) = √ (82) =
Step 8..
Step 3. Know the difference between perfect and imperfect squares
Until now, the solutions to our square root operations have been nice clean integers. This is not always the case, in fact square roots can sometimes have solutions consisting of very long and uncomfortable decimals. Numbers whose square roots are whole numbers (in other words, without fractions or decimals) are called perfect squares. All the examples listed above (9, 25 and 64) are perfect squares because when you extract their square roots, you get integers (3, 5 and 8).
Conversely, numbers that do not give integers as a result when the square root is extracted are called imperfect squares. Extracting the square root of one of these numbers usually results in a fraction or decimal number. Sometimes, the decimals involved can be somewhat complicated. For example √ (13) = 3, 605551275464…
Step 4. Memorize the first 10-12 perfect squares
As you have probably noticed, extracting the square root of perfect squares can be quite easy! Since solving these problems is very simple, it is worth taking some time to memorize the square roots of the first ten perfect squares. You will have a lot to do with these numbers, so by taking the time to memorize them you can save yourself a lot later. The first 12 perfect squares are:
-
12 = 1 × 1 =
Step 1.
-
22 = 2 × 2 =
Step 4.
-
32 = 3 × 3 =
Step 9.
-
42 = 4 × 4 =
Step 16.
-
52 = 5 × 5 =
Step 25.
- 62 = 6 × 6 = 36
- 72 = 7 × 7 = 49
- 82 = 8 × 8 = 64
- 92 = 9 × 9 = 81
- 102 = 10 × 10 = 100
- 112 = 11 × 11 = 121
- 122 = 12 × 12 = 144
Step 5. Simplify the square roots by removing perfect squares whenever possible
Finding the square roots of imperfect squares can be quite tricky at times, especially if you're not using a calculator (you'll find some tricks to make the process easier in the section below). However, it is often possible to simplify the numbers under the root and make them easier to do the calculations. To do this, you simply have to factor the number under the root, take the square root of each factor which is a perfect square, and write the solution out of the radical. It's definitely easier than it looks - read on to find out more!
- Let's say we want to find the square root of 900. At first glance it seems pretty difficult! However, it won't be that complicated if we factor 900 into factors. Factors are the numbers that can be multiplied together to form another number. For example, since you can get 6 by multiplying 1 × 6 and 2 × 3, the factors of 6 are 1, 2, 3 and 6.
- Instead of doing the math with the number 900, which is quite complicated, write it as 9 × 100. Now, since 9, which is a perfect square, is separated by 100, we can extract its square root individually. √ (9 × 100) = √ (9) × √ (100) = 3 × √ (100). In other words, √ (900) = 3√(100).
-
We can therefore simplify it further by decomposing 100 into the factors 25 and 4. √ (100) = √ (25 × 4) = √ (25) × √ (4) = 5 × 2 = 10. Therefore we can say that √ (900) = 3 (10) =
Step 30..
Step 6. Use imaginary numbers for the square roots of negative numbers
Think about it: what number multiplied by itself gives -16? Neither 4 nor -4: squaring them you get in both cases the positive number 16. Do you give up? In fact, there is no way to write the square root of -16 (and any other negative number) with real numbers. In these cases, imaginary numbers (usually in the form of letters or symbols) must be used to substitute them for the square root of the negative number. For example, the variable i is usually used for the square root of -1. As a general rule, the square root of a negative number will always be (or will include) an imaginary number.
Note that although imaginary numbers cannot be represented with classic digits, they can still be treated like real numbers in many respects. For example, the square roots of negative numbers can be squared to get those same negative numbers, just like any other square root of a positive number. For example, i 2 = - 1.
Part 2 of 3: Using the Column Division Method
Step 1. Arrange the square root as in a column division
Although it may take quite a while, this method allows you to solve the square roots of rather difficult imperfect squares without the use of a calculator. To do this, we'll use a solving method (or algorithm) that is similar, but not exactly identical, to basic column splitting.
- Start by writing the square root in the same form as a column division. For example, let's say we want to find the square root of 6.45, which is definitely not a convenient perfect square. First, write the usual root symbol (√) and the number below it. Then, make a line under the number so that it comes into a sort of little "box", like a division by column. When finished, you should have a long tailed "√" symbol and a 6.45 written underneath.
- Write the numbers above the root to make sure you leave space.
Step 2. Group the digits in pairs
To begin to solve the problem, group the digits of the number under the sign of the radical in pairs, starting with the decimal point. It may be useful to make small marks (such as periods, bars, commas, etc.) between the various pairs to keep track of them.
In our example, we will divide 6.45 like this: 6-, 45-00. Note the presence of a number "advancing" on the left, that's okay.
Step 3. Find the largest number whose square is less than or equal to the first "group" of digits
Start with the first number, the first pair on the left. Choose the largest number with a square that is less than or equal to that "group" of digits. For example, if the group of digits was 37, choose 6, because 62 = 36 <37 but 72 = 49> 37. Write this number above the first group. It is the first digit of your solution.
-
In our example, the first group of 6-, 45-00 is made up of 6. The largest number that squared is less than or equal to 6 is
Step 2., since 22 = 4. We write a "2" above the 6 present under the root.
Step 4. Double the number you just typed, bring it down and subtract it
Take the first digit of your solution (the number you just found) and double it. Write it under the first group and subtract it to find the difference. Bring the next pair of numbers underneath next to the result. Finally, write on the left the last digit of the double (of the first digit) of the solution and leave a space next to it.
In our example, we will start by taking double 2, the first digit of our solution. 2 × 2 = 4. So, we'll subtract 4 from 6 (our first "group"), getting 2 as the result. Next, we will bring down the next group (45) to get 245. Finally, we will write 4 again on the left, leaving a small space to write in, like this: 4_
Step 5. Fill in the blank
Next, you will need to add a digit on the right side of the number you just wrote on the left. Choose the largest possible figure (to multiply by the new number), but still less than or equal to the number you "brought down". For example, if the number you "brought down" is 1700 and the number on the left is 40_, you will need to fill in the blank with "4" because 404 × 4 = 1616 <1700, while 405 × 5 = 2025. The number that you find at this point of the procedure, it will be the second digit of your solution, and you can then add it above the root sign.
-
In our example, we need to find the number that filling the blank with 4_ × _ gives the greatest possible result - but still less than or equal to 245. In this case, the answer will be
Step 5.. 45 × 5 = 225, while 46 × 6 = 276.
Step 6. Continue, using the "blank" numbers for the result
Continue to perform this modified column division method until you start getting zeros by subtracting from the numbers "below", or until you reach the level of approximation required. When you are done, the numbers you used in each step to fill in the blanks (plus the very first number) will form the digits of your solution.
-
Continuing our example, we subtract 225 from 245 to get 20. Then, we bring down the next pair of digits, 00, to make 2000. By doubling the numbers above the root sign, we get 25 × 2 = 50. Solving the white space of 50_ × _ = / <2000, we get
Step 3.. At this point, we will have "253" above the root sign. By repeating the same process one more time, we will get 9 as the next digit.
Step 7. Move above the decimal point from your starting "dividend"
To complete your solution, you will need to put the decimal point in the right place. Luckily, it's easy: all you have to do is match it with the decimal point of the starting number. For example, if the number under the root sign is 49, 8, you will simply have to move the comma between the two numbers above 9 and 8.
In our example, the number under the root sign is 6.45, so we'll just move the comma above by putting it between digits 2 and 5 of our result, getting 2, 539.
Part 3 of 3: Quickly Perform an Approximate Estimate of Imperfect Squares
Step 1. Find non-perfect squares by making rough estimates
Once you have memorized the perfect squares, finding the square roots of the imperfect squares will become much easier. Since you already know more than a dozen perfect squares, any number that is between two of these can be found by "smoothing" more and more a rough estimate between these values. To begin with, find the two perfect squares between which the number is located. Next, determine which of these two numbers comes closest.
For example, let's say we need to find the square root of 40. Since we have memorized the perfect squares, we can say that 40 is between 62 and 72, ie between 36 and 49. Since 40 is greater than 62, its square root will be greater than 6; and since it is less than 72, its square root will also be less than 7. Also, 40 is a little closer to 36 than 49, so the result will likely be closer to 6 than 7. In the next steps, we will further refine the accuracy of our solution.
Step 2. Approximate the square root to one decimal place
Once you have found two perfect squares between which the number lies, it will become a simple matter of increasing your approximation until you reach a solution that satisfies you; the more you go into detail, the more accurate the solution will be. To begin, choose a decimal place "of the value of tenths" for the solution, it does not have to be exact, but it will save you a lot of time using common sense to choose the one that comes closest to the right result.
In our example problem, a reasonable approximation for the square root of 40 could be 6, 4, as we know, from the above procedure, that the solution is probably closer to 6 than to 7.
Step 3. Multiply the approximate number by itself
Then square your estimate. Unless you're really lucky, you won't get the starting number right away - you'll be slightly above or below it. If your solution is a slightly higher number than given, try again with a slightly lower approximation (and vice versa if the solution is lower, try with a higher estimate).
- Multiply 6.4 by itself to get 6.4 × 6.4 = 40, 96, which is slightly greater than the starting number we want to find the root of.
- Then, as we have gone beyond the required result, we will multiply the number by itself by one-tenth less than our overestimate, yielding 6.3 × 6.3 = 39, 69, which this time is slightly less than the starting number. This means that the square root of 40 is somewhere between 6, 3 and 6, 4. Also, since 39.69 is closer to 40 than 40.96, we will know that the square root will be closer to 6.3 than 6.4.
Step 4. Continue the approximation process as required
At this point, if you are satisfied with the solutions found, you may want to simply pick and use one as a rough estimate. If you want to get a more accurate solution, all you have to do is choose an estimate for the "cents" figure that brings this approximation between the first two. By continuing with this method, you will be able to get three decimal places for your solution, and even four, five and so on, it will just depend on how much detail you want to get.
