Square Root of 54

The square root is the inverse of the square of the number. 54 is not a perfect square. The square root of 54 is expressed in both radical and exponential form.

In the radical form, it is expressed as √54, whereas (54)^(1/2) in the exponential form. √54 ≈ 7.34847, 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 54 is broken down into its prime factors:

Step 1: Finding the prime factors of 54 Breaking it down, we get 2 x 3 x 3 x 3: 2^1 x 3^3

Step 2: Now we found out the prime factors of 54. The second step is to make pairs of those prime factors. Since 54 is not a perfect square, therefore the digits of the number can’t be grouped in pairs completely.

Therefore, calculating √54 using prime factorization gives us √(2 x 3^2 x 3) = 3√6.

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 the case of 54, we need to group it as 54.

Step 2: Now we need to find n whose square is less than or equal to 54. The closest perfect square number is 49, which gives n as 7. Now the quotient is 7 and the remainder is 54 - 49 = 5.

Step 3: Add a decimal point to the quotient and bring down a pair of zeros to make the dividend 500.

Step 4: The new divisor is 14 (2 times the quotient 7) followed by a digit, say x, such that 14x * x ≤ 500. The number is 3, so 143 * 3 = 429.

Step 5: Subtract 429 from 500, the difference is 71.

Step 6: Bring down another pair of zeros, making it 7100.

Step 7: Repeat the process to refine the value of x. Continue doing these steps until you obtain a satisfactory approximation.

So the square root of √54 ≈ 7.348.

Approximation method is another method for finding the 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 54 using the approximation method.

Step 1: Now we have to find the closest perfect square of √54. The smallest perfect square less than 54 is 49, and the largest perfect square greater than 54 is 64. √54 falls somewhere between 7 and 8.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (54 - 49) / (64 - 49) = 5 / 15 = 0.3333.

Using the formula we identified the decimal point of our square root estimate. The next step is adding the value we got initially to the decimal number which is 7 + 0.3333 = 7.3333, so the square root of 54 is approximately 7.348.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16 and the inverse of the square is the square root, which 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. That is the reason it is also known as a principal square root.
   
• Prime factorization: Expressing a number as the product of its prime factors is known as prime factorization.
   
• Long division method: A method used to find square roots of non-perfect square numbers by dividing and averaging over several steps to reach an approximate value.