Square Root of 1.09

The square root is the inverse of the square of the number. 1.09 is not a perfect square. The square root of 1.09 is expressed in both radical and exponential form. In the radical form, it is expressed as √1.09, whereas (1.09)^(1/2) in the exponential form. √1.09 = 1.04403, 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 long-division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
   
• Long division method
   
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 1.09 is broken down into its prime factors.

Step 1: Finding the prime factors of 1.09 Since 1.09 is not a perfect square or an integer, the prime factorization method is not applicable.

Therefore, calculating 1.09 using prime factorization is not possible.

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 this case, we consider 109 (ignoring the decimal initially).

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

Step 3: Bring down 09 which becomes 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 2n. Now we find n such that 2n×n is less than or equal to 09. Step 5: Consider n as 0, then 2×0×0 = 0.

Step 6: Subtract 0 from 9, the remainder is 9, and the quotient is 10.

Step 7: Since the dividend is less than the divisor, add a decimal point. Add two zeroes to the dividend. The new dividend is 900.

Step 8: Find the new divisor, which is 204, because 204×4 = 816.

Step 9: Subtract 816 from 900 to get the result 84.

Step 10: Now the quotient is 1.04.

Step 11: Continue with these steps until two numbers after the decimal point are obtained.

So the square root of √1.09 is approximately 1.044.

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 1.09 using the approximation method.

Step 1: Find the closest perfect squares of √1.09. The smallest perfect square close to 1.09 is 1, and the largest is 1.21. √1.09 falls between 1 and 1.1.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Largest perfect square - smallest perfect square). Using the formula (1.09 - 1) / (1.21 - 1) = 0.45 Add the value obtained to the smaller square root, which is 1 + 0.044 = 1.044, so the square root of 1.09 is approximately 1.044.

• Square root: The square root is the inverse of squaring a number. Example: 3² = 9, and the square root of 9 is √9 = 3.

• Irrational number: An irrational number cannot be written in the form of p/q, where q is not equal to zero and p and q are integers. Example: √2, π.

• Decimal: A decimal is a number that includes a whole part and a fractional part separated by a decimal point. Example: 1.09, 3.14.

• Long division method: A mathematical method for dividing large numbers to find the quotient and remainder, often used to find square roots of non-perfect squares.

• Approximation method: A technique for estimating the value of a number or expression, commonly used to find square roots of non-perfect squares.