Square Root of 3529

The square root is the inverse of the square of the number. 3529 is not a perfect square. The square root of 3529 is expressed in both radical and exponential form. In the radical form, it is expressed as √3529, whereas (3529)^(1/2) in the exponential form. √3529 ≈ 59.3986, which is an irrational number because it cannot be expressed as 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 gives the prime factorization of a number. Now let us look at how 3529 is broken down into its prime factors.

Step 1: Finding the prime factors of 3529 Breaking it down, we get 11 x 13 x 13 x 19: 11¹ x 13² x 19¹

Step 2: Now that we found the prime factors of 3529, the second step is to pair those prime factors. Since 3529 is not a perfect square, the digits of the number can’t be grouped entirely into pairs. Therefore, calculating 3529 using prime factorization is not straightforward.

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 3529, we need to group it as 29 and 35.

Step 2: Now, we need to find n whose square is 35. We can say n is '5' because 5 x 5 = 25 is less than or equal to 35. The quotient is 5, and after subtracting 25 from 35, the remainder is 10.

Step 3: Now let us bring down 29, which becomes the new dividend. Add the old divisor with the same number 5 + 5 to get 10, which will be our new divisor.

Step 4: The new divisor is now 10n. We need to find the value of n such that 10n x n ≤ 1029. Let us consider n as 9, as 109 x 9 = 981.

Step 5: Subtract 981 from 1029, resulting in a remainder of 48.

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 zeros to the dividend. Now the new dividend is 4800.

Step 7: The new divisor will be 598, as 598 x 8 = 4784.

Step 8: Subtract 4784 from 4800 to get 16.

Step 9: The quotient now is 59.3.

Step 10: Continue repeating these steps until we get two numbers after the decimal point. The square root of √3529 ≈ 59.40.

The approximation method is another way to find 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 3529 using the approximation method.

Step 1: Identify the closest perfect squares around 3529. The smallest perfect square less than 3529 is 3481 (59²), and the largest perfect square greater than 3529 is 3600 (60²). √3529 falls between 59 and 60.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula (3529 - 3481) ÷ (3600 - 3481) = 48 ÷ 119 ≈ 0.403 Add this value to the lower perfect square root: 59 + 0.403 ≈ 59.403. So the square root of 3529 is approximately 59.40.

• Square root: The square root is the inverse operation of squaring a number. For example, if 4² = 16, then the square root of 16 is √16 = 4.

• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction, with a numerator and denominator that are integers and the denominator not zero.

• Perfect square: A perfect square is a number that can be expressed as the product of an integer with itself, e.g., 4, 9, 16, 25.

• Decimal: A decimal is a number that includes a fractional part, expressed with a decimal point, e.g., 7.86, 8.65, 9.42.

• Principal square root: A number has both positive and negative square roots, but the positive square root is typically referred to as the principal square root.