Square Root of 10.7

The square root is the inverse of the square of a number. 10.7 is not a perfect square. The square root of 10.7 is expressed in both radical and exponential forms. In radical form, it is expressed as √10.7, whereas (10.7)^(1/2) in exponential form. √10.7 ≈ 3.2711, 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 prime factorization method is not used; instead, the long-division and approximation methods are used. Let us now learn about the following methods:

• Long division method
• Approximation method

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 10.7, we consider it as 10.70.

Step 2: Now, find n whose square is less than or equal to 10. The nearest perfect square is 9, which is 3^2. The quotient is 3, and after subtracting 9 from 10, the remainder is 1.

Step 3: Bring down the next pair of digits (70), making the new dividend 170. Double the current quotient (3) to get 6 and use it as the start of the new divisor.

Step 4: Find n such that 6n × n ≤ 170. The largest digit satisfying this condition is 2, since 62 × 2 = 124.

Step 5: Subtract 124 from 170, getting a remainder of 46.

Step 6: Add a decimal point to the quotient and bring down two zeroes to the remainder, making it 4600.

Step 7: Double the current quotient (32) to get 64, and use it as the start of the new divisor.

Step 8: Find n such that 64n × n ≤ 4600. We find that n = 7 since 647 × 7 = 4529.

Step 9: Subtract 4529 from 4600, leaving a remainder of 71.

Step 10: Repeat the process until the desired precision is achieved.

Thus, the square root of 10.7 is approximately 3.2711.

Approximation is another method for finding square roots; it is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 10.7 using the approximation method.

Step 1: Identify the closest perfect squares around 10.7. The closest perfect squares are 9 (3^2) and 16 (4^2). √10.7 falls between 3 and 4.

Step 2: Now, apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Using this, (10.7 - 9) / (16 - 9) = 1.7 / 7 ≈ 0.243. Add this to the smaller square root: 3 + 0.243 = 3.243.

Thus, the approximate square root of 10.7 is around 3.2711 when refined.

• Square root: A square root is the operation of finding a number that, when multiplied by itself, gives the original number. Example: 3^2 = 9, and the inverse is √9 = 3.
   
• Irrational number: An irrational number cannot be written as a simple fraction; it cannot be expressed as a ratio of two integers.
   
• Principal square root: Of the two square roots (positive and negative) of a number, the positive one is known as the principal square root.
   
• Decimal: A number that includes a fractional part separated from the integer part by a decimal point, such as 3.2711.
   
• Long division method: A technique for finding the square root of a number by dividing the number into pairs of digits and estimating the root digit by digit.