Square Root of 1.5

The square root is the inverse of the square of the number. 1.5 is not a perfect square. The square root of 1.5 is expressed in both radical and exponential form. In the radical form, it is expressed as √1.5, whereas (1.5)^(1/2) in the exponential form. √1.5 ≈ 1.2247, 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 and approximation methods 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 1.5 is broken down into its prime factors.

Step 1: Finding the prime factors of 1.5 Breaking 1.5 into a fraction, we have 3/2. The prime factorization of 3 is 3, and for 2 is 2.

Step 2: Since 1.5 is not a perfect square, we cannot pair the prime factors completely.

Thus, calculating 1.5 using prime factorization directly for square roots is not feasible.

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: Start by grouping the numbers from right to left. Here, we consider 15 (1.5 without the decimal).

Step 2: Find n whose square is less than or equal to 1. For n = 1, 1 × 1 = 1. Subtracting, we get 0.

Step 3: Bring down 50 (from 1.5, treating it as 150). The new dividend is 50.

Step 4: Double the divisor (1), resulting in 2. Now, find a digit x such that 2x × x ≤ 50.

Step 5: For x = 2, 22 × 2 = 44. Subtract 44 from 50, leaving a remainder of 6.

Step 6: Since the dividend is less than the divisor, add a decimal point and bring down two zeros, making the new dividend 600.

Step 7: Double the current quotient (12), yielding 24. Find a new digit y such that 24y × y ≤ 600.

Step 8: For y = 2, 242 × 2 = 484. Subtract 484 from 600, resulting in 116.

Step 9: Continue this process to achieve the desired precision.

The square root of 1.5 is approximately 1.2247.

The approximation method is an easy method for finding the square roots of a given number. Now let us learn how to find the square root of 1.5 using the approximation method.

Step 1: Identify the smallest and largest perfect squares around 1.5. The smallest perfect square is 1 (√1 = 1), and the largest is 4 (√4 = 2). √1.5 falls between 1 and 2.

Step 2: Apply interpolation between the perfect squares. (1.5 - 1) / (4 - 1) = 0.5 / 3 ≈ 0.1667.

Step 3: Approximate: 1 + 0.1667 = 1.1667.

Step 4: Refine as needed.

The approximation result approaches 1.2247 via further iterations.

• 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 the positive square root that has more prominence due to its uses in the real world. This is why it is also known as the principal square root.
   
• Decimal: A decimal number contains a whole number and a fractional part separated by a decimal point, for example, 1.5, 2.75, and 3.6.
   
• Long Division Method: This is a method used to find the square root of non-perfect squares, involving a series of division steps to achieve an accurate result.