Square Root of 1134

The square root is the inverse of the square of the number. 1134 is not a perfect square. The square root of 1134 is expressed in both radical and exponential form. In the radical form, it is expressed as √1134, whereas (1134)^(1/2) in the exponential form. √1134 ≈ 33.661, 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 1134 is broken down into its prime factors:

Step 1: Finding the prime factors of 1134 Breaking it down, we get 2 x 3 x 3 x 3 x 3 x 7: (2^1 times 3^4 times 7^1)

Step 2: Now we found out the prime factors of 1134. The second step is to make pairs of those prime factors. Since 1134 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 1134 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 1134, we need to group it as 34 and 11.

Step 2: Now we need to find n whose square is less than or equal to 11. We can say n as ‘3’ because 3 × 3 = 9 is lesser than or equal to 11. Now the quotient is 3, and after subtracting 11 - 9, the remainder is 2.

Step 3: Now let us bring down 34, which is the new dividend. Add the old divisor with the same number 3 + 3 = 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 234. Let's consider n as 3, now 63 × 3 = 189.

Step 6: Subtract 234 from 189; the difference is 45, and the quotient is 33.

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 4500.

Step 8: Now we need to find the new divisor, which is 336, because 336 × 9 = 3024.

Step 9: Subtracting 3024 from 4500 gives us the result 1476.

Step 10: Now the quotient is 33.6.

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 √1134 is approximately 33.66.

The 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 1134 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1134. The smallest perfect square of 1134 is 1024, and the largest perfect square of 1134 is 1156. √1134 falls somewhere between 32 and 34.

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 (1134 - 1024) / (1156 - 1024) ≈ 0.91. 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 32 + 0.91 = 32.91, so the square root of 1134 is approximately 33.66.

• 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, it is always a 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: Prime factorization is breaking down a number into its basic prime number factors. For example, the prime factorization of 18 is 2 × 3 × 3.

• 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.