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How to Solve for Square Root Without Calculator

To solve for the square root of a number without using a calculator, you can use an estimate and then refine it. Start by estimating the square root, then divide the number by your estimated answer. This will give you a better estimate to work with.

You can repeat this process until you get close enough to the correct answer. Another method is to break down the perfect squares that make up your original number into their factors before finding its square root. To find which perfect squares are factors of your original number, list out all of its prime factors and calculate how many times each one appears in that list (called its exponent).

Then look up what numbers multiplied together create those perfect squares and work out their roots separately before combining them back together again as if they were one big factor tree equation – using multiplication and addition/subtraction only – until you arrive at the final answer.

  • Step 1: Identify the square root number
  • This is the number you want to find the square root of
  • Step 2: Determine if it’s a perfect square
  • A perfect square is a number that is equal to an integer multiplied by itself, such as 4×4 or 9×9
  • If your number isn’t a perfect square, skip ahead to Step 5
  • Step 3: Find two numbers whose product equals your original number and are close together in value (such as 3 x 6 = 18)
  • These two numbers will become the factors for finding your answer
  • Step 4: Use addition and subtraction on these two numbers until one side becomes 0 (for example, subtracting 3 from 6 leaves you with 3 on one side and 0 on the other)
  • The remaining non-zero factor is your answer
  • In this case, since 6 minus 3 equals 3, then the root of 18 would be 3
  • Step 5: Estimate what half of your original number might be if it isn’t a perfect square (for example 22 has 11 as its approximate halfway point)
  • Then use trial-and-error methods to determine which pair of integers multiply together closest to that starting point without going over it (in this case 10 x 12 = 120)
  • Subtract any extra amount you went over from half of your original number and divide by whatever factor was used in multiplication; this gives you an estimate for solving non-perfect squares roots too!
How to Solve for Square Root Without Calculator


How Do You Manually Calculate Square Roots?

Calculating the square root of a number manually need not be a daunting task, as it is often thought to be. The process can actually be quite simple once you understand the underlying principles. To calculate the square root of a number, first start by writing down two factors that when multiplied together equal your given number.

For example, if you are trying to find the square root of 25, then 5 x 5 = 25. Next, separate each factor into two parts such that one part is double (or more) than the other; in this case we could write 5 = 3 + 2 or 5 = 4 + 1. Now add those numbers together and divide them by two; in our example 3 + 2 / 2 = 2.5 or 4+1/2=2.5 – either way we end up with an answer of 2.5 for √25 .

This same technique works for most whole numbers but will become increasingly difficult as larger numbers are used due to both precision and time constraints; in these cases there may not exist any factors which can easily be separated into two parts and added together so other methods must be employed such as long division or Newton’s method..

What is the Formula for a Square Root?

The formula for a square root is an expression that finds the number which, when multiplied by itself, produces a given number. In mathematical terms, it is written as √x = y, where x is the given number and y is the unknown value of its square root. To calculate this equation manually, one must use long division or guess-and-check strategies; however, calculators are now available to quickly compute these types of equations.

The most common way to express square roots in mathematics is with radical notation—the symbol “√” followed by the given value inside a pair of parentheses with an index indicating which root you wish to find: For example, if you wanted to find the fourth root of 81 (4√81), then your answer would be 3 since 32 = 9 and 42 = 16.

How Do You Solve Square Roots Easily?

Square roots can be a tricky concept to understand and solve, but the good news is that there are several easy methods you can use to calculate them. First and foremost, if you have a calculator handy, this is often the easiest way to solve square root problems. Simply enter the number into your calculator, hit the “square root” button (typically denoted by ‘√’), and voila!

You’ll get your answer instantly. Another popular method for solving square roots is called prime factorization – which involves breaking down a number into its prime factors or smallest possible components and then using those components as factors of the original number in order to find its square root. Lastly, if neither of these options are available to you and all else fails, sometimes trial-and-error can help too; start with an estimate of what you think might be close enough to the correct answer (e.g., 10) and then keep adjusting it until your guess comes close enough for accuracy’s sake!

How to Calculate Square Root Without Calculator

How to Find Square Root Manually

Finding the square root of a number manually can be done with a process known as long division. To start, you will need to divide the number into two-digit numbers and group them together in pairs from right to left. Then, you must determine the largest perfect square that fits inside each pair and write it down above that pair.

Next, subtract this squared number from the first pair and bring down the next two-digit number to form your new dividend. By repeating these steps until all digits are divided, you will end up with your desired answer -the square root of your original number!

How to Find Square Root Without Calculator Using Prime Factorization

Finding the square root of a number without using a calculator can be done by prime factorization. This process involves breaking down the number into its prime factors, then taking out pairs of equal factors from each side of the equation until only one is left in the middle. For instance, to find the square root of 16, you would need to break it down into its prime factors (2x2x2x2), and then take out two 2s from each side (2×2).

The resulting answer would be 2, so √16 = 2.

How to Calculate Square Root by Division Method

The Division Method is an easy and efficient way to calculate the square root of a number. This method involves dividing the number into two parts, one containing the digits before the decimal point and one containing those after it. After that, you divide each part by 2 until both halves are single digit numbers.

The outputs from this process can be multiplied together to give you a good approximation of the square root of your original number.

How to Find Square Root With Calculator

Finding the square root of a number is an important skill that can be easily done with a calculator. All modern scientific calculators have a square root function, usually denoted by √. To find the square root of any number, simply enter the value in your calculator and press the ‘√’ button to get the result.

For example, if you wanted to find out what 4² was equal to, you would type ‘4’ into your calculator followed by pressing the ‘√’ button and it will return 2 as your answer.


This blog post has provided a helpful and easy-to-follow guide to solving for square root without using a calculator. By following this three step process, anyone can easily solve for the square root of any number. With practice, it can become second nature to calculate the square root of numbers by hand.

Now that you have learned how to solve for square roots by hand, you will no longer need to rely on calculators in order to figure out this common mathematical equation!