Square Root of 901

The square root is the inverse of the square of a number. 901 is not a perfect square. The square root of 901 is expressed in both radical and exponential form. In the radical form, it is expressed as √901, whereas in exponential form it is (901)^(1/2). √901 ≈ 30.0333, which is an irrational number because it cannot be expressed as a fraction 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 is the prime factorization of a number. Now let us look at how 901 is broken down into its prime factors:

Step 1: Finding the prime factors of 901 Breaking it down, we get 17 x 53: 17^1 x 53^1

Step 2: Now we have found the prime factors of 901. The second step is to make pairs of those prime factors. Since 901 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating the square root of 901 using prime factorization is not straightforward.

The long division method is particularly useful 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 901, we group it as 01 and 90.

Step 2: Now we need to find n whose square is less than or equal to 90. We use n as ‘9’ because 9 x 9 = 81, which is less than 90. The quotient is 9, and after subtracting 90 - 81, the remainder is 9.

Step 3: Now let us bring down 01, making the new dividend 901. Add the old divisor with the same number: 9 + 9 = 18, which will be our new divisor.

Step 4: The new divisor will be the sum of the old divisor and the quotient. Now we get 18n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 18n x n ≤ 901. Let's consider n as 4, now 18 x 4 x 4 = 288.

Step 6: Subtract 901 - 288; the difference is 613, and the quotient becomes 30.

Step 7: Since the dividend is greater than the divisor, we need to add a decimal point to continue accurately. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 61300.

Step 8: Now we need to find the new divisor, which is 180 because 180 x 3 = 540.

Step 9: Subtracting 540 from 613, we get the result 73.

Step 10: Now the quotient is 30.03.

Step 11: Continue these steps until we get two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.

So the square root of √901 is approximately 30.03.

The approximation method is another method for finding square roots. It is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 901 using the approximation method.

Step 1: Now we have to find the closest perfect square to √901.

The smallest perfect square less than 901 is 900, and the largest perfect square greater than 901 is 961.

√901 falls somewhere between 30 and 31.

Step 2: Now we need to apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (901 - 900) ÷ (961 - 900) = 1/61 ≈ 0.0164.

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 30 + 0.0164 ≈ 30.03, so the square root of 901 is approximately 30.03.

• Square root: A square root is the inverse of a square. Example: 4² = 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.
   
• Long division method: A method used to find the square root of non-perfect squares by approximating the root using long division.
   
• Perfect square: A perfect square is a number that can be expressed as the product of an integer with itself. Example: 4, 9, 16 are perfect squares.
   
• Approximation method: A method used to estimate the square root of a number by identifying the nearest perfect squares and calculating the difference.