Square Root of 1029

The square root is the inverse of the square of the number. 1029 is not a perfect square. The square root of 1029 is expressed in both radical and exponential form. In the radical form, it is expressed as √1029, whereas \(1029^{1/2}\) is in exponential form. √1029 ≈ 32.085, 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 1029 is broken down into its prime factors:

Step 1: Finding the prime factors of 1029 Breaking it down, we get 3 x 3 x 3 x 7 x 17: \(3^3 \times 7 \times 17\)

Step 2: Now we found out the prime factors of 1029. The second step is to make pairs of those prime factors. Since 1029 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating √1029 using prime factorization is not feasible for finding an exact whole number.

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 1029, we need to group it as 29 and 10.

Step 2: Now we need to find n whose square is closest to 10. We can say n as ‘3’ because \(3 \times 3 = 9\) is lesser than or equal to 10. Now the quotient is 3; after subtracting 9 from 10, the remainder is 1.

Step 3: Now let us bring down 29, 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 \(6n \times n \leq 129\). Let us consider n as 2; now \(6 \times 2 \times 2 = 124\).

Step 6: Subtract 124 from 129; the difference is 5, 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 500.

Step 8: Now we need to find the new divisor that is 642 because \(642 \times 1 = 642\).

Step 9: Subtracting 642 from 5000, we continue the process further for more precision.

Step 10: 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 √1029 is approximately 32.085.

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 1029 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √1029. The smallest perfect square less than 1029 is 1024 (which is \(32^2\)), and the largest perfect square greater than 1029 is 1089 (which is \(33^2\)). √1029 falls somewhere between 32 and 33.

Step 2: Now we need to apply the formula that is \((\text{Given number} - \text{smallest perfect square}) / (\text{Greater perfect square} - \text{smallest perfect square})\). Going by the formula \((1029 - 1024) / (1089 - 1024) = 5/65 = 0.077\). 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 \(32 + 0.077 \approx 32.085\).

So the square root of 1029 is approximately 32.085.

• 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 the principal square root.
   
• Long division method: A step-by-step approach to finding the square root of a number by dividing it into pairs of digits and solving iteratively.
   
• Approximation method: A technique used to estimate the square root of a number by finding nearby perfect squares and calculating the decimal portion based on their proximity.