Square Root of 10201

The square root is the inverse of the square of the number. 10201 is a perfect square. The square root of 10201 is expressed in both radical and exponential form. In radical form, it is expressed as √10201, whereas (10201)^(1/2) in exponential form. √10201 = 101, which is a rational number because it can 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, such as 10201. For non-perfect square numbers, methods like long-division and approximation 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 10201 is broken down into its prime factors.

Step 1: Finding the prime factors of 10201

Breaking it down, we get 101 x 101: 101^2

Step 2: Now we found out the prime factors of 10201. Since 10201 is a perfect square, the digits of the number can be grouped in pairs. Therefore, calculating 10201 using prime factorization gives us √10201 = 101.

The long division method is useful for both perfect and 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 10201, we need to group it as 01, 20, and 10.

Step 2: Now we need to find n whose square is less than or equal to 10. We can use n = 3 because 3 x 3 = 9, which is less than 10. The quotient is 3, and the remainder is 1.

Step 3: Bring down the next pair 20 to make it 120. Add the old divisor with the same number 3 + 3 to get 6, which will be our new divisor.

Step 4: The new divisor will be placed beside the dividend to make it 6n as the new divisor. We need to find the value of n.

Step 5: The next step is finding 6n x n ≤ 120; let us consider n as 2, so 62 x 2 = 124, which is too large. Try n = 1, so 61 x 1 = 61.

Step 6: Subtract 120 - 61, and the difference is 59, and the quotient is 31.

Step 7: Bring down the next pair 01 to make it 5901. Add the old divisor with the same number 61 + 1 to get 62, which will be our new divisor.

Step 8: The next step is finding 62n x n ≤ 5901; let us consider n as 9, so 629 x 9 = 5661.

Step 9: Subtract 5901 - 5661, and the difference is 240, and the quotient is 310.

Step 10: 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 24000.

Step 11: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero.

So the square root of √10201 is 101.

The approximation method 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 10201 using the approximation method.

Step 1: Now we have to find the closest perfect square of √10201. The closest perfect square of 10201 is 10000 and 10404. √10201 falls between 100 and 102.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (10201 - 10000) / (10404 - 10000) = 201 / 404 = 0.4975.

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 100 + 0.4975 ≈ 100.5, so the square root of 10201 is approximately 101, which we know is exact by other methods.

• Square root: A square root is the inverse of a square. Example: 10^2 = 100, and the inverse of the square is the square root that is √100 = 10.
   
• Perfect square: A perfect square is a number that has a whole number as its square root. For example, 16 is a perfect square because √16 = 4.
   
• Rational number: A rational number is a number that can be expressed in the form of p/q, where q is not equal to zero, and p and q are integers.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors. For example, 10201 = 101 x 101.
   
• Long division method: A method used to find the square root of numbers by dividing the number into groups and using division steps to reach the square root.