Square Root of 10.9

The square root is the inverse of the square of the number. 10.9 is not a perfect square. The square root of 10.9 is expressed in both radical and exponential form. In the radical form, it is expressed as √10.9, whereas (10.9)^(1/2) in the exponential form. √10.9 ≈ 3.3015, 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:

• Prime factorization method
• Long division method
• Approximation method

The prime factorization method is not suitable for non-perfect squares like 10.9. Therefore, calculating √10.9 using prime factorization is not possible as it cannot be broken down into integer prime factors.

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: Begin by grouping the numbers from right to left. In the case of 10.9, treat it as 1090 after adding a decimal point and zeros.

Step 2: Find a number n whose square is less than or equal to 10. The closest is 3 because 3 × 3 = 9. Subtract 9 from 10 to get a remainder of 1.

Step 3: Bring down a pair of zeros to make the new dividend 100.

Step 4: Double the divisor (3) to get 6 and determine the next digit of the quotient such that 60n × n is less than or equal to 100. The suitable n is 1 as 61 × 1 = 61.

Step 5: Subtract 61 from 100 to get a remainder of 39. Bring down another pair of zeros to make it 3900.

Step 6: Continue the process to get the next digit of the quotient and repeat the steps until the desired precision is reached.

The square root of √10.9 is approximately 3.3015.

The approximation method is another way to find square roots. It is easy and useful for non-perfect squares such as 10.9.

Step 1: Find the closest perfect squares around √10.9. The closest perfect squares are 9 and 16. √10.9 falls between √9 (3) and √16 (4).

Step 2: Use linear approximation.

Since 10.9 is closer to 9 than to 16, calculate the approximate decimal part by interpolation. (10.9 - 9) / (16 - 9) ≈ 0.2714

Add this to 3, giving approximately 3.2714.

Refine further to get 3.3015.

• 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.
   
• Decimal approximation: An approximate value of a number expressed in decimal form, often used when the exact value is irrational.
   
• Principal square root: A number has both positive and negative square roots; however, the positive square root is often used due to its practical applications, known as the principal square root.
   
• Interpolation: A method of estimating unknown values that fall between known values, often used in approximation methods.