Square Root of 536

The square root is the inverse of the square of a number. 536 is not a perfect square. The square root of 536 is expressed in both radical and exponential form. In radical form, it is expressed as √536, whereas in exponential form it is expressed as (536)^(1/2). √536 ≈ 23.1517, 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 often 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 536 is broken down into its prime factors:

Step 1: Finding the prime factors of 536 Breaking it down, we get 2 x 2 x 2 x 67: 2³ x 67¹

Step 2: Now that we have found the prime factors of 536, the second step is to make pairs of those prime factors. Since 536 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating the square root of 536 using prime factorization directly 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: To begin with, we need to group the numbers from right to left. In the case of 536, we group it as 36 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 2^2 = 4, which is less than 5. Now the quotient is 2, and subtracting 4 from 5 gives a remainder of 1.

Step 3: Bring down the next pair, 36, making the new dividend 136. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor will be 4n, and we need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 136. Let us consider n as 3; now 43 × 3 = 129.

Step 6: Subtract 129 from 136; the difference is 7, 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 zeros to the dividend. Now the new dividend is 700.

Step 8: Find the new divisor: 466, as 466 × 1 = 466.

Step 9: Subtracting 466 from 700 gives 234.

Step 10: Continue these steps until we get two numbers after the decimal point. The square root of √536 is approximately 23.1517.

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 536 using the approximation method.

Step 1: Find the closest perfect squares to √536. The smallest perfect square less than 536 is 529 (23²), and the largest perfect square greater than 536 is 576 (24²). √536 falls between 23 and 24.

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

Using the formula: (536 - 529) ÷ (576 - 529) = 7 ÷ 47 ≈ 0.1489.

Adding the value obtained to the whole number: 23 + 0.1489 ≈ 23.1517.

• Square root: A square root is the inverse operation of squaring a number. For example, 4² = 16, and the inverse of the square is the square root, so √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.

• Principal square root: A number has both positive and negative square roots; however, the positive square root is more commonly used due to its applications in the real world.

• Prime factorization: The process of expressing a number as the product of its prime factors.

• Decimal: A number that includes a whole number and a fractional part, separated by a decimal point, such as 7.86 or 8.65.