Square Root of 15/2

The square root is the inverse of the square of the number. 15/2 is not a perfect square. The square root of 15/2 is expressed in both radical and exponential form. In the radical form, it is expressed as, √(15/2), whereas (15/2)^(1/2) in the exponential form. √(15/2) = 1.93649, 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 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. For the fraction 15/2, we will find the prime factors of the numerator and the denominator separately.

Step 1: Finding the prime factors of 15 and 2 Breaking it down, we get 15 = 3 x 5 and 2 = 2. So the prime factorization of 15/2 is 3 x 5 / 2.

Step 2: Since 15/2 is not a perfect square, calculating √(15/2) using prime factorization directly is not possible, but understanding the factorization helps in other methods.

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. Here is how to find the square root using the long division method, step by step.

Step 1: To begin with, express 15/2 as a decimal, which is 7.5.

Step 2: Now, group the number 7.5 from right to left. For this, consider it as 7.50.

Step 3: Find n such that n^2 is less than or equal to 7. The nearest perfect square is 4 (2 x 2), so n is 2. Subtract 4 from 7 to get 3, and bring down 50 to make it 350.

Step 4: Double the divisor (2) to make it 4. Find the largest digit "d" such that 4d x d is less than or equal to 350. The result is 46 x 6 = 276.

Step 5: Subtract 276 from 350 to get 74, and bring down two zeros to make it 7400.

Step 6: Continue this process to get the square root to desired decimal places.

So the square root of √(15/2) is approximately 1.936.

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 15/2 using the approximation method.

Step 1: Convert 15/2 to a decimal, which is 7.5.

Step 2: Find the closest perfect squares around 7.5. The nearest perfect squares are 4 (2^2) and 9 (3^2). √(7.5) falls between 2 and 3.

Step 3: Apply the approximation formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Using the formula, (7.5 - 4) / (9 - 4) = 0.7.

Step 4: Add the initial whole number, 2 + 0.7 = 2.7. Since this is an approximation to the nearest whole number, refine it to get a more accurate result. Thus, the square root of 15/2 is approximately 1.936.

• Square root: A square root is the inverse of a square. For example, 4^2 = 16 and the inverse of the square is the square root, that is, √16 = 4.

• 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.

• Fraction: A fraction consists of a numerator and a denominator, such as 15/2, where 15 is the numerator and 2 is the denominator.

• Decimal: If a number has a whole number and a fraction in a single number, it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.

• Approximation: Approximating means finding a value that is close to the actual value. Often used when exact values are not feasible, especially with irrational numbers.