In the days before calculators were invented, students and professors had to calculate square roots manually. Several different ways have been developed to overcome this difficult process. Some ways give a rough estimate and others give an exact value. To learn how to find the square root of a number using just simple operations, see Step 1 below to get started.
Step
Method 1 of 2: Using Prime Factorization
Step 1. Divide your number into perfect square factors
This method uses the factors of a number to find the square root of the number (depending on the number, the answer can be an exact number or a close approximation). The factors of a number are a set of other numbers which, when multiplied, produce that number. For example, you could say that the factors of 8 are 2 and 4 because 2 × 4 = 8. Meanwhile, perfect squares are whole numbers that are the product of other whole numbers. For example, 25, 36, and 49 are perfect squares because they are 5. respectively2, 62, and 72. As you might have guessed, perfect square factors are factors that are also perfect squares. To start finding the square root through prime factorization, first try to simplify your number to its perfect square factors.
- Let's use an example. We want to find the square root of 400 manually. To start, we'll divide the number into its perfect square factors. Since 400 is a multiple of 100, we know that 400 is divisible by 25 – a perfect square. With a quick division of the shadow, we find that 400 divided by 25 equals 16. Coincidentally, 16 is also a perfect square. Thus, the perfect square factors of 400 are 25 and 16 because 25 × 16 = 400.
- We can write it as: Sqrt(400) = Sqrt(25 × 16)
Step 2. Find the square root of your perfect square factors
The multiplication property of the square root states that for any number a and b, Sqrt(a × b) = Sqrt(a) × Sqrt(b). Because of this property, now, we can now find the square root of our perfect square factors and multiply them to get our answer.
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In our example, we will find the square roots of 25 and 16. See below:
- Root(25 × 16)
- Root(25) × Root(16)
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5 × 4 =
Step 20.
Step 3. If your number cannot be factored perfectly, simplify your answer to its simplest form
In real life, often the numbers you need to find the square root of aren't pleasant whole numbers with obvious perfect square factors like 400. In these cases, it's possible that we can't find the right answer. as a whole number. However, by finding as many perfect square factors as you can find, you can find the answer in terms of a square root that is smaller, simpler, and easier to calculate. To do this, reduce your number to a combination of perfect square factors and imperfect square factors, then simplify.
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Let's use the square root of 147 as an example. 147 is not a product of two perfect squares, so we can't get the exact integer value as above. However, 147 is the product of one perfect square and another number – 49 and 3. We can use this information to write our answer in its simplest form as follows:
- Root(147)
- = Root(49 × 3)
- = Sqrt(49) × Sqrt(3)
- = 7 × Root(3)
Step 4. If needed, estimate
With your square root in its simplest form, it's usually fairly easy to get a rough estimate of the number answer by guessing the value of the remaining square root and multiplying it. One way to guide your guess is to look for perfect squares that are greater than and less than the number in your square root. You'll notice that the decimal value of the number in your square root is between the two numbers, so you can guess the value between the two numbers.
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Let's return to our example. because 22 = 4 and 12 = 1, we know that Root(3) is between 1 and 2 – probably closer to 2 than 1. We estimate 1, 7. 7 × 1, 7 = 11, 9. If we check our answer on the calculator, we can see that our answer is quite close to the real answer which is 12, 13.
This also applies to larger numbers. For example, Root(35) can be approximated between 5 and 6 (possibly closer to 6). 52 = 25 and 62 = 36. 35 is between 25 and 36, so the square root must be between 5 and 6. Since 35 is only one less than 36, we can say with confidence that the square root is slightly less than 6. Checking with a calculator will give us the answer is about 5, 92 – we are right.
Step 5. Alternatively, reduce your number to its least common factors as your first step
Finding the factors of perfect squares isn't necessary if you can easily determine the prime factors of a number (factors that are also prime numbers). Write your number in terms of its least common factors. Then, find the pairs of prime numbers that match your factors. When you find two prime factors that are the same, remove these two numbers from the square root and place one of these numbers outside the square root.
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For example, find the square root of 45 using this method. We know that 45 × 5 and we know that under 9 = 3 × 3. Thus, we can write our square root in terms of the factors like this: Sqrt(3 × 3 × 5). Just remove both 3s and put one 3 outside the square root to simplify your square root to its simplest form: (3)Root(5).
From here, we will be easy to estimate.
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As a final example problem, let's try to find the square root of 88:
- Root(88)
- = Root(2 × 44)
- = Root(2 × 4 × 11)
- = Root(2 × 2 × 2 × 11). We have some 2 in our square root. Since 2 is a prime number, we can remove a pair of 2s and put one of them outside the square root.
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= Our square root in its simplest form is (2) Sqrt(2 × 11) or (2) Root(2) Root(11).
From here, we can estimate Sqrt(2) and Sqrt(11) and find the approximate answer as we want.
Method 2 of 2: Finding the Square Root Manually
Using the Long Division Algorithm
Step 1. Separate the digits of your number into pairs
This method uses a process similar to long division to find the exact square root digit by digit. While not mandatory, you may find it easier to carry out this process if you visually organize your workplace and your numbers into easy-to-work parts. First, draw a vertical line dividing your work area into two sections, then draw a shorter horizontal line near the top right to divide the right section into a smaller top section and a larger bottom section. Next, separate your digits into pairs, starting at the decimal point. For example, following this rule, 79,520,789,182, 47897 becomes "7 95 20 78 91 82. 47 89 70". Write your number at the top left.
For example, let's try to calculate the square root of 780, 14. Draw two lines to divide your workplace as above and write "7 80. 14" in the upper left. It doesn't matter if the leftmost number is a single number, and not a pair of numbers. You will write your answer (square root 780, 14) at the top right
Step 2. Find the largest integer whose square value is less than or equal to the leftmost number (or pair of numbers)
Start at the far left of your number, both number pairs and single numbers. Find the largest perfect square that is less than or equal to this number, then find the square root of this perfect square. This number is n. Write n in the upper right and write the square of n in the lower right quadrant.
In our example, the far left is the number 7. Because we know that 22 = 4 ≤ 7 < 32 = 9, we can say that n = 2 because 2 is the largest integer whose square value is less than or equal to 7. Write 2 in the upper right quadrant. This is the first digit of our answer. Write 4 (square value of 2) in the lower right quadrant. This number is important for the next step.
Step 3. Subtract the number you just calculated from the leftmost pair
Like long division, the next step is to subtract the value of the square we just found from the part we just analyzed. Write this number under the first part and subtract it, writing your answer below it.
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In our example, we'll write 4 under 7, then subtract it. This subtraction yields an answer
Step 3..
Step 4. Drop the next pair
Move down the next section of the number for which you are looking for the square root, next to the subtraction value you just found. Next, multiply the number in the upper right quadrant by two and write the answer in the lower right quadrant. Next to the number you just wrote down, leave a space for the multiplication problem you will do in the next step by writing '"_×_="'.
In our example, the next pair of our numbers is "80". Write "80" next to 3 in the left quadrant. Next, multiply the number at the top right by two. This number is 2, so 2 × 2 = 4. Write "'4"' in the lower right quadrant, followed by _×_=.
Step 5. Fill in the blanks in the right quadrant
You must fill in all the blanks you just wrote in the right quadrant with the same whole number. This integer must be the largest integer that makes the product in the right quadrant less than or equal to the number currently on the left.
In our example, we fill in the blanks with 8, resulting in 4(8) × 8 = 48 × 8 = 384. This value is greater than 384. Thus, 8 is too large, but 7 might work. Write 7 in the blanks and solve: 4(7) × 7 = 329. 7 is a correct number because 329 is less than 380. Write 7 in the upper right quadrant. This is the second digit in the square root of 780, 14
Step 6. Subtract the number you just calculated from the number now on the left
Continue with the subtraction chain using the long division method. Take the product of the problem in the right quadrant and subtract it from the number that is now on the left, while writing your answers below.
In our example, we will subtract 329 from 380, which gives the result 51.
Step 7. Repeat step 4
Derive the next part of the number for which you are looking for the square root. When you reach the decimal point in your number, write the decimal point in your answer in the upper right quadrant. Then, multiply the number in the upper right by 2 and write it next to the blank multiplication problem ("_ × _") as above.
In our example, since we are now dealing with the decimal point in 780, 14, write the decimal point after our current answer in the upper right. Next, lower down the next pair (14) in the left quadrant. Twice the number in the upper right (27) equals 54, so write "54 _×_=" in the lower right quadrant
Step 8. Repeat steps 5 and 6
Find the largest digit to fill in the blanks on the right, which gives an answer less than or equal to the number currently on the left. Then, solve the problem.
In our example, 549 × 9 = 4941, which is less than or equal to the number on the left (5114). 549 × 10 = 5490 is too big, so 9 is your answer. Write 9 as the next digit in the upper right quadrant and subtract the product from the number on the left: 5114 minus 4941 equals 173
Step 9. To continue counting the digits, lower the pair of zeros on the left, and repeat steps 4, 5, and 6
For greater accuracy, continue this process to find the hundreds, thousands, and more places in your answer. Continue using this cycle until you find the decimal place you want.
Understanding the Process
Step 1. Imagine the number you calculated the square root of as the area S of a square
Since the area of a square is P2 where P is the length of one of the sides, then by trying to find the square root of your number, you are actually trying to calculate the length P of that side of the square.
Step 2. Determine the letter variables for each digit of your answer
Set the variable A as the first digit of P (the square root we are trying to calculate). B will be the second digit, C the third digit, and so on.
Step 3. Determine the letter variables for each part of your starting number
Set variable Sa for the first pair of digits in S (your initial value), Sb for the second pair of digits, etc.
Step 4. Understand the relationship between this method and long division
This method of finding the square root is basically a long division problem that divides your initial number by the square root, giving you the square root of the answer. Just like in the long division problem, you're only interested in the next digit in each step. In this way, you are only interested in the next two digits in each step (which is the next digit in each step for the square root).
Step 5. Find the largest number whose square value is less than or equal to Sa.
The first digit of A in our answer is the largest integer whose square value does not exceed Sa (ie A so that A² Sa < (A+1)²). In our example, Sa = 7, and 2² 7 < 3², so A = 2.
Note that, for example, if you wanted to divide 88962 by 7 using long division, the first steps are pretty much the same: you'll see the first digit of 88962 (which is 8) and you're looking for the largest digit which, when multiplied by 7, is less than or equal to 8 Basically, you're looking for d so that 7×d 8 < 7×(d+1). In this case, d will be equal to 1
Step 6. Imagine the value of the square whose area you are about to start working on
Your answer, the square root of your starting number, is P, which describes the length of the square with area S (your starting number). Your grades for A, B, C, represent the digits in the value of P. Another way of saying this is 10A + B = P (for a two-digit answer), 100A + 10B + C = P (for a three-digit answer), etc.
In our example, (10A+B)² = P2 = S = 100A² + 2×10A×B + B². Remember that 10A+B represents our answer, P, with B in the ones position and A in the tens position. For example, with A=1 and B=2, then 10A+B equals 12. (10A+B)² is the total area of the square, while 100A² is the area of the largest square in it, B² is the area of the smallest square in it, and 10A×B is the area of the two remaining rectangles. By carrying out this long and convoluted process, we find the total area of a square by adding up the areas of the squares and rectangles inside.
Step 7. Subtract A² from Sa.
Decrease one pair of digits (Sb) of S. Value of Sa Sb close to the total area of the square, which you just used to subtract the larger inner square. The remainder can be thought of as the number N1, which we got in step 4 (N1 = 380 in our example). N1 equals 2×:10A×B + B² (area of two rectangles plus the area of the smaller square).
Step 8. Find N1 = 2×10A×B + B², which is also written as N1 = (2×10A + B) × B
In our example, you already know N1 (380) and A(2), so you have to find B. B is most likely not a whole number, so you really need to find the largest integer B such that (2×10A + B) × B N1. So you have: N1 < (2×10A + (B+1)) × (B+1).)
Step 9. Finish
To solve this equation, multiply A by 2, shift the result to the tens position (the equivalent of multiplying by 10), put B in the ones position, and multiply the number by B. In other words, solve (2×10A + B) × B. This is exactly what you do when you write "N_×_=" (with N=2×A) in the lower right quadrant in step 4. In step 5, you find the largest integer B that corresponds to the number below it so that (2× 10A + B) × B N1.
Step 10. Subtract the area (2×10A + B) × B from the total area
This subtraction results in the area S-(10A+B)² that has not been calculated (and which will be used to calculate the next digit in the same way).
Step 11. To calculate the next digit, C, repeat the process
Lower the next pair (Sc) of S to get N2 on the left, and find the largest C so that you have (2×10×(10A+B)+C) × C N2 (equivalent to writing twice the two-digit number "AB" followed by "_× _=". Find the largest matching digit in the blanks, which gives an answer less than or equal to N2, as before.
Tips
- Moving a decimal point by a multiple of two digits in a number (a multiple of 100), means moving a decimal point by a multiple of one digit in its square root (a multiple of 10).
- In this example, 1.73 can be considered a "remainder": 780, 14 = 27, 9² + 1.73.
- This method can be used for any base, not just base 10 (decimal).
- You can use calculus which is more convenient for you. Some people write the result above the initial number.
- An alternative way of using repeated fractions is to follow this formula: z = (x^2+y) = x + y/(2x + y/(2x + y/(2x + …))). For example, to calculate the square root of 780, 14, the integer whose squared value is closest to 780, 14 is 28, so z=780, 14, x=28, and y=-3, 86. Entering values and calculating estimates only for x + y/(2x) it yields (in simplest terms) 78207/20800 or about 27, 931(1); next term, 4374188/156607 or approximately 27, 930986(5). Each term adds about 3 decimal places to the accuracy of the previous number of decimal places.
