Square Root of 534

The square root is the inverse of the square of the number. 534 is not a perfect square. The square root of 534 is expressed in both radical and exponential form. In the radical form, it is expressed as √534, whereas (534)^{(1/2) i}n the exponential form. √534 ≈ 23.110, 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: -

1. Prime factorization method 
2. Long division method 
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 534 is broken down into its prime factors.

Step 1: Finding the prime factors of 534 Breaking it down, we get 2 x 3 x 89: ²¹ x 3¹ x 89^1.

Step 2: Now we found out the prime factors of 534. The second step is to make pairs of those prime factors. Since 534 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √534 using prime factorization is impossible.

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 534, we need to group it as 34 and 5.

Step 2: Now we need to find n whose square is less than or equal to 5. We can say n is ‘2’ because 2 x 2 = 4, which is less than 5. The quotient is 2, and after subtracting 5 - 4, the remainder is 1.

Step 3: Now let us bring down 34, making it the new dividend, 134. Add the old divisor with the same number 2 + 2 to get 4, which will be our new provisional divisor.

Step 4: The new divisor will be 4 followed by a digit we need to find. We need to find n such that 4n x n ≤ 134.

Step 5: The next step is finding 4n x n ≤ 134. Let us consider n as 3, now 43 x 3 = 129.

Step 6: Subtract 134 from 129; the difference is 5, and the quotient is 23.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 500.

Step 8: Now we need to find the new divisor, which is 461 because 461 x 1 = 461.

Step 9: Subtracting 461 from 500, we get the result 39.

Step 10: Now the quotient is 23.1. Step 11: Continue doing these steps until we get two numbers after the decimal point.

Suppose if there are no decimal values, continue till the remainder is zero. So the square root of √534 ≈ 23.11.

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 us learn how to find the square root of 534 using the approximation method.

Step 1: Now we have to find the closest perfect square to √534. The smallest perfect square less than 534 is 529, and the largest perfect square greater than 534 is 576. √534 falls somewhere between 23 and 24.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula, (534 - 529) / (576 - 529) = 5 / 47 ≈ 0.106.

Adding this value to 23 gives us 23 + 0.106 = 23.106, so the square root of 534 is approximately 23.11.

• 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.

• Perfect square: A perfect square is an integer that is the square of an integer. For example, 16 is a perfect square because it is 4²

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

• Long division method: A technique used to divide large numbers and also to find the square roots of non-perfect squares through a systematic process.