Square Root of 105

The square root is the inverse of the square of the number. 105 is not a perfect square. The square root of 105 is expressed in both radical and exponential form.

In the radical form, it is expressed as √105, whereas (105)^(1/2) in the exponential form. √105 ≈ 10.24695, 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, the prime factorization method is not used for non-perfect square numbers where 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 105 is broken down into its prime factors.

Step 1: Finding the prime factors of 105. Breaking it down, we get 3 x 5 x 7: 3¹ x 5¹ x 7¹

Step 2: Now we found out the prime factors of 105. The second step is to make pairs of those prime factors. Since 105 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating √105 using prime factorization is impossible.

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 105, we need to group it as 05 and 1.

Step 2: Now we need to find n whose square is 1. We can say n is ‘1’ because 1 x 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 05, which is the new dividend. Add the old divisor with the same number, 1 + 1, we 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, we need to find the value of n.

Step 5: The next step is finding 2n x n ≤ 05. Since 2 x 1 x 1 = 2, it's not possible. We continue with 2 x 0 x 0 = 0.

Step 6: Subtract 05 from 0, the difference is 5, and the quotient is 10.

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 500.

Step 8: Now we need to find the new divisor that is 20 because 202 x 2 = 404. Step 9: Subtracting 404 from 500, we get the result 96.

Step 10: Now the quotient is 10.2.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √105 is approximately 10.25.

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 105 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √105. The smallest perfect square less than 105 is 100, and the largest perfect square greater than 105 is 121. √105 falls somewhere between 10 and 11.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula, (105 - 100) ÷ (121 - 100) = 5/21 ≈ 0.238.

Using the 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 + 0.238 = 10.238, so the square root of 105 is approximately 10.238.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, that is √16 = 4.

• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.

• Principal square root: A number has both positive and negative square roots; however, it is always a positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.

• Prime factorization: The process of breaking down a number into its basic building blocks of prime numbers. For example, the prime factorization of 105 is 3 x 5 x 7.

• Long division method: A method used to find the square root of non-perfect square numbers by dividing the number into pairs and calculating step by step.