How To Calculate A Square Root By Hand With Pictures

Let’s use an example. We want to find the square root of 400 by hand. To begin, we divide the number into perfect square factors. Since 400 is a multiple of 100, we know that it’s evenly divisible by 25 - a perfect square. Quick mental division lets us know that 25 goes into 400 16 times. 16, coincidentally, is also a perfect square. Thus, the perfect square factors of 400 are 25 and 16 because 25 × 16 = 400. We would write this as: Sqrt(400) = Sqrt(25 × 16) You can also try multiplying different numbers by themselves and see if they give you the correct answer. Let’s say you were trying to find the square root of 81—you could multiply 7 by 7 to get 49, which is too low. You could go a little higher and multiply 10 by 10 to get 100, which is too high. You could then go a little lower and multiply 9 by 9. This would give you 81, making 9 your square root.

In our example, we would take the square roots of 25 and 16. See below: Sqrt(25 × 16) Sqrt(25) × Sqrt(16) 5 × 4 = 20

Let’s use the square root of 147 as an example. 147 isn’t the product of two perfect squares, so we can’t get an exact integer value as above. However, it is the product of one perfect square and another number - 49 and 3. We can use this information to write our answer in simplest terms as follows: Sqrt(147) = Sqrt(49 × 3) = Sqrt(49) × Sqrt(3) = 7 × Sqrt(3)

Let’s return to our example. Since 22 = 4 and 12 = 1, we know that Sqrt(3) is between 1 and 2 - probably closer to 2 than to 1. We’ll estimate 1. 7. 7 × 1. 7 = 11. 9 If we check our work in a calculator, we can see that we’re fairly close to the actual answer of 12. 13. This works for larger numbers as well. For example, Sqrt(35) can be estimated to be between 5 and 6 (probably very close to 6). 52 = 25 and 62 = 36. 35 is between 25 and 36, so its square root must be between 5 and 6. Since 35 is just one away from 36, we can say with confidence that its square root is just lower than 6. Checking with a calculator gives us an answer of about 5. 92 - we were right.

This works for larger numbers as well. For example, Sqrt(35) can be estimated to be between 5 and 6 (probably very close to 6). 52 = 25 and 62 = 36. 35 is between 25 and 36, so its square root must be between 5 and 6. Since 35 is just one away from 36, we can say with confidence that its square root is just lower than 6. Checking with a calculator gives us an answer of about 5. 92 - we were right.

As an example, let’s find the square root of 45 using this method. We know that 45 = 9 × 5 and we know that 9 = 3 × 3. Thus, we can write our square root in terms of its factors like this: Sqrt(3 × 3 × 5). Simply remove the 3’s and put one 3 outside the square root to get your square root in simplest terms: (3)Sqrt(5). From here, it’s simple to estimate. As one final example problem, let’s try to find the square root of 88: Sqrt(88) = Sqrt(2 × 44) = Sqrt(2 × 4 × 11) = Sqrt(2 × 2 × 2 × 11). We have several 2’s in our square root. Since 2 is a prime number, we can remove a pair and put one outside the square root. = Our square root in simplest terms is (2) Sqrt(2 × 11) or (2) Sqrt(2) Sqrt(11). From here, we can estimate Sqrt(2) and Sqrt(11) and find an approximate answer if we wish.

As an example, let’s try calculating the square root of 780. 14. Draw two lines to divide your workspace as above and write “7 80. 14” at the top of the left space. It’s O. K. that the leftmost chunk is a lone number, rather than a pair of numbers. You will write your answer (the square root of 780. 14. ) in the top right space.

In our example, the leftmost “chunk” is the number 7. Since we know that 22 = 4 ≤ 7 < 32 = 9, we can say that n = 2 because it’s the largest integer whose square is less than or equal to 7. Write 2 in the top right quadrant. This is the first digit of our answer. Write 4 (the square of 2) in the bottom right quadrant. This number will be important in the next step.

In our example, we would write 4 below 7, then subtract. This gives us an answer of 3.

In our example, the next pair in our number is “80”. Write “80” next to the 3 in the left quadrant. Next, multiply the number in the top right by two. This number is 2, so 2 × 2 = 4. Write “‘4”’ in the bottom right quadrant, followed by ×=.

In our example, filling in the blank spaces with 8, gives us 4(8) × 8 = 48 × 8 = 384. This is greater than 380. Therefore, 8 is too big, but 7 will probably work. Write 7 in the blank spaces and solve: 4(7) × 7 = 329. 7 checks out because 329 is less than 380. Write 7 in the top right quadrant. This is the second digit in the square root of 780. 14.

In our example, we would subtract 329 from 380, which gives us 51.

In our example, since we are now encountering the decimal point in 780. 14, write a decimal point after our current answer the top right. Next, drop the next pair (14) down in the left quadrant. Two times the number on the top right (27) is 54, so write “54 ×=” in the bottom right quadrant.

In our example, 549 × 9 = 4941, which is lower than or equal to the number on the left (5114). 549 × 10 = 5490, which is too high, so 9 is our answer. Write 9 as the next digit in the top right quadrant and subtract the result of the multiplication from the number on the left: 5114 minus 4941 is 173.

Note that, for instance, if you wanted to divide 88962 by 7 via long division, the first step would be similar: you would be looking at the first digit of 88962 (8) and you would want the biggest digit that, when multiplied by 7, is lower than or equal to 8. Essentially, you’re finding d so that 7×d ≤ 8 < 7×(d+1). In this case, d would be equal to 1.

In our example, (10A+B)² = L2 = S = 100A² + 2×10A×B + B². Remember that 10A+B represents our answer L with B in the units position and A in the tens position. For instance, with A=1 and B=2, 10A+B is simply the number 12. (10A+B)² is the area of the whole square, while 100A² the area of the biggest square inside, B² is the area of the smallest square, and 10A×B is the area of each of the two remaining rectangles. By performing this long, convoluted process, we find the area of the entire square by adding up the areas of the squares and rectangles inside it.