One of the benefits of maths is that there are usually several ways to find the answer to a problem. In the instance of finding the square root of a number, you can always use the handy square root button on a calculator. However, there are also various ways that you can work out the answer yourself when all you have is a pen and paper.
Long division is one of the best ways to find square roots, especially if you are working with a large number. Although it is helpful to have prior knowledge of factors and square numbers, you can use this method without requiring too many complex mathematical skills.
Many people are intimidated by long division at first but find it easier to understand when they realize that it is a systematic and logical approach that can help you break down and work with larger numbers in a simplified method.
In this guide, we'll look at the best way to find the square root of a number using long division, as well as the other methods that can help you find the same answer.
The square root is the factor of a number that can be multiplied by itself to get the original number. For example, the square root of 16 is 4 as 4 x 4 = 16. This would be written as √16 = 4. The square root symbol is called a radical, whilst the number that sits under it is called a radicand.
There are many reasons to find the square root of a number. They are commonly used in finance to work out the rate of return, as well as lengths and distances, and the radius of circles.
The rate of return on assets can be calculated using the formula R = √(V2 / V0) – 1. R is the annual rate of return, V0 is the value at the start, and V2 is the value of the asset after two years. For example, if your asset was worth £100 to begin with, and you eventually sell it for £150, V0 = £100 and V2 = £150. The equation would be as follows:
- R = √(150 / 100) – 1
- R = √(1.5) – 1
- R = 1.2 – 1
- R = 0.2
In this instance, 0.2 means an annual return of 20%, so the asset returned 20% annually across the two years.
Many different industries, including engineering, architecture and construction, use the Pythagorean Theorem to find the distance and lengths of a triangle's sides. The formula for finding the sides on a right angle (90 degrees) triangle is a2 + b2 = c2. To find the hypotenuse (the triangle's longest side), you can take the square root of the two other sides using the equation √(a2 + b2) = c.
For example, if a triangle has a side 5cm in length and another with a length of 7cm, the equation would be:
a = 5 and b = 7
- √(a2 + b2) = c
- √(52 + 72) = c
- √(25 + 49) = c
- √(74) = c
- 8.6 = c
Therefore, the triangle's hypotenuse is 8.6cm.
Division is one of four primary operations in mathematics, along with addition, subtraction and multiplication. You can use long division to calculate the square root of a large number if you don't have a calculator handy.
The first step is to write the number that you are trying to find the square root of beneath a horizontal line and to the right of a vertical line. However, rather than writing as one number, you should separate it into multiple pairs so that you can work with smaller numbers. For example, if the starting number is 6,084, the paired digits would be 60 and 84. Think of the largest perfect square number that you can that is equal to or just less than the two paired digits. In the case of the example number, the largest perfect square that is under 60 is 49. You will then write the square root of 49 (which is 7).
Next, write a 7 above the line that 60 sits below. 7 x 7 = 49, so write 49 directly below 60. You must then add the two sevens together, which leaves you with 14. Write this number to the left of the table, leaving a space for your next digit. Subtract 49 from 60, which leaves you with 11. The remaining pair is 84, so write this number next to the 11, which leaves you with the total number 1,184.
Use trial and error to find the two products of 1,184. The last digit is 4, which means that the last digit of the two factors must end in 4 as well. The two single-digit prime factors that end in a 4 when multiplied together are 2 x 2 = 4 and 8 x 8 = 64. However, in this instance, the answer is 8 because when you write it in the space, you left next to the 14, it will give you 148 x 8 = 1,184. Write an 8 next to the 7 that you had previously written above the 60. This will give you the final answer of 78.
Add 148 to 8, which will give you 156. The result you get here will always be double your final answer, which is seen here as 78 + 78 = 156. The final answer is √6084 = 78.
While the long division method is a good way to find a large number's square root if you are working it out by hand, there are other ways that you can find the same answer. There are a number of online calculators that let you input the number and will supply you with the square root.
You can also find the factors of the given number and then separate the factors into their separate square roots. Following this, you can find the squares of the individual factors. With these simplified squares, you can multiply them together to find your final answer. For example:
√225 can be simplified to √25 and √9 as 25 x 9 = 225. The square root of 25 is 5, and the square root of 9 is 3, so 5 x 3 = 15. Therefore, the square root of 225 is 15.
In some situations, you won't necessarily know which factors of a square root are squared. In this instance, you can find the most obvious factor of a number and work from that. For example, a clear factor of 225 is 5, so you could use the equation √225 = √(5 x 45). The next step is to find the factors of 45, which are √(5 x 5 x 9). The final factor to simplify is 9, which equals 3 x 3. The long-form factor would therefore look like √(5 x 5 x 3 x 3). Although the numbers are listed multiple times, you only need to include them as a factor once. This will leave you with the equation 5 x 3 = 15.
Estimating is one of the best ways to help you calculate the square root of a number if you are unsure about its factors. This is especially helpful if the number you are working with doesn't have a perfect square number. For example, you may be trying to find the square root of 10. Your first estimate may be that 4 is a factor of 10. You would need to perform the following calculations:
- Divide the starting number with your estimate. In this case, it would be 10 ÷ 4 = 2.5
- Next, divide your estimated number by the previous answer. So, 4 + 2.5 = 6.5
- Finally, you need to half that answer, which would give you 6.5 ÷ 2 = 3.25
The answer 3.25 is your new estimate, which you will use in the previous calculations in place of the 4. You can keep repeating these steps until you reach 10 or are as close as you can get when you square the final answer. For example, if you repeat the steps twice more, you will get 3.1623. 3.16232 = 10.00014
While these calculations can help you to find a number's square root, the long division method is easier when you have a larger number as you don't have to estimate the factors.
Finding the square roots of numbers can be helpful for various reasons, such as calculating the rate of return on assets or finding the lengths and distances of land for construction. Calculators tend to come with a square root button which easily allows you to find the square root of a number, no matter if it is a perfect square number or not.
One of the most reliable ways to solve square root problems when working with larger numbers is to use long division. This involves separating out the given number into pairs so that you can find the largest perfect square numbers that fit into them. Another method that is fairly easy is to estimate the given number's factors. This is easier if you have a smaller number. However, you may have to repeat the process several times to find the closest answer, especially if your original estimate was far off.