Square Root of 1.56

The square root is the inverse of the square of the number. 1.56 is not a perfect square. The square root of 1.56 is expressed in both radical and exponential form. In the radical form, it is expressed as √1.56, whereas (1.56)^(1/2) in the exponential form. √1.56 ≈ 1.249, 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 1.56 is broken down into its prime factors. Since 1.56 is not a whole number, prime factorization in the traditional sense cannot be directly applied. Instead, we would generally use other methods such as the long division method or approximation to find the square root of 1.56.

The long division method is particularly used for non-perfect square numbers. Let's learn how to find the square root of 1.56 using the long division method, step by step.

Step 1: Begin by grouping the digits of 1.56 from right to left. Here, we don't have to group since it's a decimal number less than 10.

Step 2: Determine the largest number whose square is less than or equal to 1. We can start with 1 because 1 × 1 = 1. Subtract to get a remainder of 0.

Step 3: Bring down 56 (from 1.56) to make it a new dividend, 156.

Step 4: Double the quotient obtained (1) to get 2 and make it the new divisor, followed by a blank digit, say 'n'.

Step 5: Find the largest possible value of ‘n’ such that 2n × n is less than or equal to 156. We try n = 6, as 26 × 6 = 156.

Step 6: Subtract 156 from 156, getting a remainder of 0.

Step 7: The quotient obtained is 1.2, but since we need more precision, we add a decimal point and zeros to continue the process to get a more accurate result.

Step 8: Continue with the process to obtain more decimal places in the quotient until the desired precision is reached.

Thus, the square root of 1.56 is approximately 1.249.

The approximation method provides an easy way to estimate the square root of a given number. Let's find the square root of 1.56 using this method.

Step 1: Determine two perfect squares between which 1.56 lies. These are 1 (1²) and 4 (2²), so √1.56 is between 1 and 2.

Step 2: Use the approximation formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Using the formula: (1.56 - 1) / (4 - 1) = 0.56 / 3 ≈ 0.187

Step 3: Add this decimal to the smaller square root (1): 1 + 0.187 ≈ 1.187 This is a rough approximation. More precise calculations through methods like long division give us approximately 1.249.

• Square root: A square root is the inverse of squaring a number. For example, the square root of 4 is 2, as 2² = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction; its decimal form is non-repeating and non-terminating. For instance, √1.56 is irrational.
   
• Decimal number: A number that contains a decimal point, which separates the whole part from the fractional part, such as 1.56.
   
• Long division method: A step-by-step process to find the square root of a number by dividing, multiplying, and subtracting iteratively.
   
• Approximation method: A technique to estimate the value of a square root by comparing it to nearby perfect squares.