Square Root of 10240

The square root is the inverse of the square of a number. 10240 is not a perfect square. The square root of 10240 is expressed in both radical and exponential form. In radical form, it is expressed as √10240, whereas (10240)^(1/2) is the exponential form. √10240 ≈ 101.192885, 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; for these, 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 10240 is broken down into its prime factors.

Step 1: Finding the prime factors of 10240 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5: 2^8 x 5^2

Step 2: Now we found out the prime factors of 10240. The second step is to make pairs of those prime factors. Since 10240 is not a perfect square, the digits of the number can’t be grouped perfectly into pairs. Therefore, calculating 10240 using prime factorization directly 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 10240, we need to group it as 10, 24, and 0.

Step 2: Now we need to find n whose square is ≤ 10. We can say n is '3' because 3 x 3 = 9, which is less than or equal to 10. Now the quotient is 3; after subtracting 10 - 9, the remainder is 1.

Step 3: Now let us bring down 24, which is the new dividend. Double the old divisor (which is 3) to get 6 as the new divisor.

Step 4: The new divisor will be 6n, where n is a digit of the quotient. We need to find the largest n such that 6n × n ≤ 124. Let n = 2, then 62 × 2 = 124.

Step 5: Subtract 124 from 124, the difference is 0, and the quotient is now 32.

Step 6: Bring down the next pair of digits, which is 00.

Step 7: The new divisor becomes 64 (by doubling 32), and we need to find the largest digit n such that 64n × n ≤ 0.

Step 8: The process continues until we achieve the desired precision.

So the square root of √10240 ≈ 101.192.

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 10240 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √10240. The smallest perfect square less than 10240 is 10000 (which is 100^2), and the largest perfect square is 10404 (which is 102^2). √10240 falls somewhere between 100 and 102.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula, (10240 - 10000) ÷ (10404 - 10000) = 240 ÷ 404 ≈ 0.59406.

Using the formula, the approximation of our square root is 100 + 0.59406 = 100.59406, so the square root of 10240 is approximately 100.59406.

• 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 is more commonly used due to its applications in the real world. This is why it is also known as the principal square root.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors.
   
• Long division method: The long division method is a procedure used to find the square root of a number by systematically dividing it into groups of digits and finding the largest square.