Square Root of 1045

The square root is the inverse of the square of the number. 1045 is not a perfect square. The square root of 1045 is expressed in both radical and exponential form. In the radical form, it is expressed as √1045, whereas (1045)^(1/2) in exponential form. √1045 ≈ 32.343, 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 and approximation methods 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 1045 is broken down into its prime factors.

Step 1: Finding the prime factors of 1045 Breaking it down, we get 5 × 11 × 19.

Step 2: We found the prime factors of 1045. Since 1045 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √1045 using prime factorization directly is not possible.

The long division method is particularly used for non-perfect square numbers. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin, group the numbers from right to left. In the case of 1045, we need to group it as 10 and 45.

Step 2: Find a number whose square is less than or equal to 10. We can say this number is ‘3’ because 3 × 3 = 9. The quotient is 3, and the remainder is 1 after subtracting 9 from 10.

Step 3: Bring down the next pair, which is 45, making the new dividend 145. Add the old divisor with the same number, 3 + 3 = 6, to get the new divisor.

Step 4: The new divisor will be 6n. Find the largest value of n such that 6n × n ≤ 145. The value n is 2, so 62 × 2 = 124.

Step 5: Subtract 124 from 145, the result is 21, and the quotient is 32.

Step 6: Since the dividend is less than the divisor, add a decimal point. Add two zeroes to the dividend to make it 2100.

Step 7: The new divisor is 644. Find a number n such that 644n × n is less than or equal to 2100. Continue this process to achieve better precision.

So, the square root of √1045 is approximately 32.343.

Approximation 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 1045 using the approximation method.

Step 1: Find the closest perfect squares of √1045. The smallest perfect square less than 1045 is 1024 (32^2), and the largest perfect square greater than 1045 is 1089 (33^2). √1045 falls between 32 and 33.

Step 2: Apply the approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using (1045 - 1024) / (1089 - 1024) = 21 / 65 = 0.323.

The approximate square root is 32 + 0.323 = 32.323.

• Square root: A square root is the inverse operation of squaring a number. For example, if 4^2 = 16, then √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction; it has a non-repeating, non-terminating decimal expansion.
   
• Principal square root: The non-negative square root of a number. For example, the principal square root of 25 is 5.
   
• Factors: Numbers you can multiply together to get another number. For example, factors of 12 are 1, 2, 3, 4, 6, and 12.
   
• Perfect square: A number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.