Square Root of 107

The square root is the inverse of the square of the number. 107 is not a perfect square. The square root of 107 is expressed in both radical and exponential form.

In radical form, it is expressed as √107, whereas (107)^(1/2) in exponential form. √107 ≈ 10.34408, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, the long-division method and approximation method are used. Let us now learn the following methods: -

1. Prime factorization method 
2. Long division method 
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 107 is broken down into its prime factors:

Step 1: Finding the prime factors of 107 Breaking it down, we get 107 as a prime number itself.

Step 2: Since 107 is not a perfect square, calculating 107 using prime factorization is not possible as it cannot be grouped in pairs.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step:

Step 1: To begin with, we need to group the numbers from right to left. In the case of 107, we need to group it as 07 and 1.

Step 2: Now we need to find n whose square is 1. We can say n as ‘1’ because 1 × 1 is lesser than or equal to 1. Now the quotient is 1, and after subtracting 1-1, the remainder is 0.

Step 3: Now let us bring down 07, which is the new dividend. Add the old divisor with the same number 1 + 1 to get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 07. Let us consider n as 3; now 23 × 3 = 69, which is too large, try n as 2; 22 × 2 = 44.

Step 6: Subtract 07 from 44, and the difference is negative, so we need to re-evaluate n. Using n=2, 22 × 2 = 44 is incorrect; instead, use n=1.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 600.

Step 8: Now we need to find the new divisor. With n=3, we can check 203 × 3 = 609.

Step 9: Subtracting 609 from 600 results in a negative, refine further. Correct n for the closest approximate.

Step 10: Continue these steps until we reach two numbers after the decimal point. If no decimals are needed, continue until the remainder is zero.

So the square root of √107 ≈ 10.34

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 107 using the approximation method:

Step 1: Now we have to find the closest perfect squares to √107. The nearest perfect squares are 100 and 121. √107 falls somewhere between 10 and 11.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (107 - 100) ÷ (121 - 100) = 0.333.

Using this formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number which is 10 + 0.344 = 10.344,

so the square root of 107 is approximately 10.344.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. Example: √16 = 4.

• Irrational number: An irrational number cannot be expressed as a simple fraction. Its decimal form is non-repeating and non-terminating.

• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

• Long division method: A division method used to find the square root of non-perfect squares by iteratively guessing and refining the quotient.

• Decimal approximation: The method of approximating an irrational number to a certain number of decimal places for practical use.