Square Root of 1033

The square root is the inverse of the square of the number. 1033 is not a perfect square. The square root of 1033 is expressed in both radical and exponential form. In the radical form, it is expressed as √1033, whereas \(1033^{1/2}\) in the exponential form. √1033 ≈ 32.15598, 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, for non-perfect square numbers like 1033, the long division method and approximation method are used. Let us now learn the following methods: 

• Long division method
• Approximation method

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 1033, we need to group it as 33 and 10.

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

Step 3: Now, bring down 33, which makes the new dividend 133. Add the old divisor with the same number: \(3 + 3 = 6\), which will be our new divisor.

Step 4: The new divisor will be 6n. Now we need to find the value of n such that \(6n \times n \leq 133\). Let us consider n as 2, now \(62 \times 2 = 124\).

Step 5: Subtract \(133 - 124\); the difference is 9, and the quotient is 32.

Step 6: 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 900.

Step 7: Now we need to find the new divisor that is 644, because \(644 \times 1 = 644\).

Step 8: Subtracting 644 from 900, we get the result 256.

Step 9: Continue doing these steps until we get two numbers after the decimal point. If there is no remaining value, continue until the remainder is zero.

So the square root of √1033 ≈ 32.15598.

The approximation method is an easy method to find the square root of a given number. Now let us learn how to find the square root of 1033 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √1033. The smallest perfect square less than 1033 is 1024, and the largest perfect square greater than 1033 is 1089. √1033 falls somewhere between 32 and 33.

Step 2: Now we need to apply the formula: \(\frac{(\text{Given number} - \text{smallest perfect square})}{(\text{Greater perfect square} - \text{smallest perfect square})}\) Going by the formula, \(\frac{(1033 - 1024)}{(1089 - 1024)} = \frac{9}{65} ≈ 0.1385\). 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: \(32 + 0.1385 ≈ 32.1385\).

So the square root of 1033 is approximately 32.15598.

• 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 usually the positive square root that is used in practical applications. This is known as the principal square root.
   
• Prime number: A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing the number into pairs of digits from right to left and finding the root step by step.