Square Root of 1152

The square root is the inverse of the square of the number. 1152 is not a perfect square. The square root of 1152 is expressed in both radical and exponential form. In the radical form, it is expressed as √1152, whereas (1152)¹/² in the exponential form. √1152 = 33.9411, 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 1152 is broken down into its prime factors.

Step 1: Finding the prime factors of 1152 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 = 2⁷ x 3²

Step 2: Now we found out the prime factors of 1152. The second step is to make pairs of those prime factors. We can pair 2 x 2 x 2 and the remaining 2 cannot be paired. Therefore, we can calculate √1152 as 2³ x 3 x √2 = 32√2.

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 1152, we need to group it as 11 and 52.

Step 2: Now we need to find n whose square is 11. We can say n as '3' because 3 x 3 = 9 is less than or equal to 11. Now the quotient is 3. After subtracting 9 from 11, the remainder is 2.

Step 3: Now let us bring down 52, making it the new dividend of 252. Add the old divisor with the same number, 3 + 3, to get 6 as our new divisor.

Step 4: We need to find n such that 6n x n ≤ 252. Let us consider n as 4, then 64 x 4 = 256.

Step 5: Subtract 256 from 252, the difference is -4. Since this is not possible, we try n = 3. Now, 63 x 3 = 189.

Step 6: Subtract 189 from 252, the remainder is 63, 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 6300.

Step 8: Now we need to find the new divisor, 339, because 339 x 9 = 3051.

Step 9: Subtracting 3051 from 6300, we get the result 3249.

Step 10: Now the quotient is 33.9.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero. So the square root of √1152 is approximately 33.94.

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 1152 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √1152. The smallest perfect square less than 1152 is 1024, and the largest perfect square greater than 1152 is 1296. √1152 falls somewhere between 32 and 36.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (1152 - 1024) / (1296 - 1024) = 128 / 272 ≈ 0.47. Using the formula, we identified the decimal point of our square root. The next step is adding the initial integer value to the decimal number: 32 + 0.47 = 32.47. So, the square root of 1152 is approximately 33.94.

• 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 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 the principal square root.

• Prime factorization: Prime factorization is the process of writing a number or an expression as the product of its prime factors.

• Long division method: The long division method is used to find the square root of non-perfect squares by dividing the number into pairs and finding the closest square roots step by step.