Square Root of 13.5

The square root is the inverse of the square of the number. 13.5 is not a perfect square. The square root of 13.5 is expressed in both radical and exponential form. In the radical form, it is expressed as √13.5, whereas (13.5)^(1/2) in the exponential form. √13.5 ≈ 3.6742, 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's look at how 13.5 is broken down into its prime factors.

Step 1: Converting 13.5 to a fraction gives us 135/10, which simplifies to 27/2.

Step 2: Finding the prime factors of 27, we get 3 x 3 x 3. For 2, it is already a prime number.

Step 3: Since 13.5 is not a perfect square, the digits of the number can’t be grouped in a pair.

Therefore, calculating 13.5 using prime factorization is not straightforward.

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's 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 the case of 13.5, it is grouped as 13 and 5.

Step 2: Find n whose square is less than or equal to 13. We can say n is 3 because 3 x 3 = 9, which is less than 13. Now the quotient is 3, and after subtracting 9 from 13, the remainder is 4.

Step 3: Bring down 50 (as 13.5 is considered as 1350 for calculation purposes), making the new dividend 450. Add the old divisor with the same number 3 + 3 to get 6, which will be our new divisor.

Step 4: Find n such that 6n x n ≤ 450. Let n be 7, now 67 x 7 = 469.

Step 5: Since 469 is greater than 450, try n = 6, so 66 x 6 = 396.

Step 6: Subtract 396 from 450 to get 54. The quotient now is 3.6.

Step 7: Continue this method until you get two numbers after the decimal point. If there is no decimal, continue until the remainder is zero.

The square root of √13.5 is approximately 3.674.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Let's learn how to find the square root of 13.5 using the approximation method.

Step 1: Find the closest perfect square roots for 13.5. The smallest perfect square is 9 (3^2) and the largest is 16 (4^2). √13.5 falls between 3 and 4.

Step 2: Apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Going by the formula (13.5 - 9) / (16 - 9) = 0.642857...

Using the formula, we identified the decimal point of our square root.

The next step is adding the value we got initially to the decimal number which is 3 + 0.674 ≈ 3.674, so the square root of 13.5 is approximately 3.674.

• Square root: A square root is the inverse of a square. Example: 5^2=25 and the inverse of the square is the square root, that is √25 = 5.
   
• 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 number: A decimal number has a whole number and a fraction represented together, for example: 7.86, 8.65, and 9.42 are decimals.
   
• Prime factorization: The process of expressing a number as a product of its prime factors.
   
• Long division method: A method used to find the square root of non-perfect square numbers by dividing the number into pairs and solving systematically.