Square Root of 1026

The square root is the inverse of the square of the number. 1026 is not a perfect square. The square root of 1026 is expressed in both radical and exponential form. In the radical form, it is expressed as √1026, whereas (1026)^(1/2) in the exponential form. √1026 ≈ 32.0312, 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, methods such as long-division and approximation 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 1026 is broken down into its prime factors:

Step 1: Finding the prime factors of 1026 Breaking it down, we get 2 × 3 × 3 × 3 × 19: 2^1 × 3^3 × 19^1

Step 2: Since 1026 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1026 using prime factorization is not straightforward for finding a perfect square root.

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 1026, we need to group it as 26 and 10.

Step 2: Now we need to find n whose square is less than or equal to 10. We can say n as ‘3’ because 3 × 3 = 9 is less than 10. Now the quotient is 3, and after subtracting 9 from 10, the remainder is 1.

Step 3: Bring down 26, making it the new dividend. Add the old divisor with the same number 3 + 3 to get 6, which will be our new divisor.

Step 4: The new divisor will be 6n. We need to find the value of n.

Step 5: Find 6n × n ≤ 126. Let's consider n as 2, now 62 × 2 = 124.

Step 6: Subtract 124 from 126; the difference is 2, and the quotient is 32.

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 200.

Step 8: Find the new divisor, which is 64, because 642 × 3 = 192.

Step 9: Subtracting 192 from 200, we get the result 8.

Step 10: Now the quotient is 32.0

Step 11: Continue doing these steps until we get two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.

So the square root of √1026 is approximately 32.03.

The approximation method is another way to find 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 1026 using the approximation method.

Step 1: Find the closest perfect squares around √1026. The smallest perfect square less than 1026 is 1024, and the largest perfect square greater than 1026 is 1089. √1026 falls somewhere between 32 and 33.

Step 2: Now apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square) Using the formula (1026 - 1024) ÷ (1089 - 1024) = 2/65 ≈ 0.031 Add this to the integer part: 32 + 0.031 = 32.031

So the approximate square root of 1026 is 32.031.

• 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.
   
• Principal square root: A number has both positive and negative square roots. However, the positive square root is often used in real-world applications. This is known as the principal square root.
   
• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing and subtracting in steps to achieve the desired precision.