Square Root of 1176

The square root is the inverse of the square of the number. 1176 is not a perfect square. The square root of 1176 is expressed in both radical and exponential form. In the radical form, it is expressed as √1176, whereas (1176)^(1/2) in the exponential form. √1176 ≈ 34.29286, 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 the 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 1176 is broken down into its prime factors:

Step 1: Finding the prime factors of 1176 Breaking it down, we get 2 x 2 x 2 x 3 x 7 x 7: 2^3 x 3^1 x 7^2

Step 2: Now we found out the prime factors of 1176. The second step is to make pairs of those prime factors. Since 1176 is not a perfect square, therefore the digits of the number can’t be grouped perfectly into pairs. Therefore, calculating √1176 using prime factorization gives an approximate value.

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 1176, we need to group it as 76 and 11.

Step 2: Now we need to find n whose square is closest to or less than 11. We can say n is 3 because 3 x 3 = 9, which is less than 11. Now the quotient is 3; after subtracting 9 from 11, the remainder is 2.

Step 3: Now let us bring down 76, 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 have 6n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 6n x n ≤ 276. Let us consider n as 4; now 64 x 4 = 256.

Step 6: Subtract 256 from 276; the difference is 20, and the quotient is 34.

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 2000.

Step 8: Now we need to find the new divisor, which is 689 because 689 x 2 = 1378.

Step 9: Subtracting 1378 from 2000, we get the result 622.

Step 10: Now the quotient is 34.2

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 √1176 is approximately 34.29.

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 1176 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √1176. The smallest perfect square less than 1176 is 1156, and the largest perfect square greater than 1176 is 1225. √1176 falls somewhere between 34 and 35.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula (1176 - 1156) ÷ (1225 - 1156) = 0.29 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 34 + 0.29 = 34.29, so the square root of 1176 is approximately 34.29.

• 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, 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 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: The process of determining which prime numbers multiply together to make the original number. For example, the prime factorization of 18 is 2 x 3 x 3.

• Perfect square: A number that is the square of an integer. For example, 16 is a perfect square because it is equal to 4 x 4.