Square Root of 1762

The square root is the inverse of the square of a number. 1762 is not a perfect square. The square root of 1762 is expressed in both radical and exponential form. In radical form, it is expressed as √1762, whereas in exponential form it is expressed as (1762)^(1/2). √1762 ≈ 41.9845, 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, 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 1762 is broken down into its prime factors.

Step 1: Finding the prime factors of 1762. Breaking it down, we get 2 × 881.

Step 2: Now we found out the prime factors of 1762. The second step is to make pairs of those prime factors. Since 1762 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1762 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, we need to group the numbers from right to left. In the case of 1762, we need to group it as 62 and 17.

Step 2: Now we need to find n whose square is ≤ 17. We can say n is ‘4’ because 4 × 4 = 16, which is less than or equal to 17. Now the quotient is 4. After subtracting 16 from 17, the remainder is 1.

Step 3: Now let us bring down 62, which is the new dividend. Add the old divisor with the same number 4 + 4, we get 8, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 8n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 8n × n ≤ 162. Let us consider n as 2, now 82 × 2 = 164, which is too large. Let's try n = 1, so 81 × 1 = 81.

Step 6: Subtract 81 from 162, the difference is 81, and the quotient is 41.

Step 7: 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 8100.

Step 8: Now we need to find the new divisor that is 829 because 829 × 9 = 7461.

Step 9: Subtracting 7461 from 8100, we get the result 639.

Step 10: Now the quotient is 41.9.

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

So the square root of √1762 is approximately 41.98.

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 1762 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √1762. The smallest perfect square below 1762 is 1681, and the largest perfect square above 1762 is 1849. √1762 falls somewhere between 41 and 43.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula: (1762 - 1681) ÷ (1849 - 1681) = 81 ÷ 168 ≈ 0.482.

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 41 + 0.48 = 41.48.

So the square root of 1762 is approximately 41.48.

• 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, the positive square root is more commonly used due to its applications in the real world. This is known as the principal square root.

• Prime factorization: The expression of a number as the product of its prime factors. For example, the prime factorization of 1762 is 2 × 881.

• Long division method: A technique used to find the square root of numbers that are not perfect squares by dividing the number step-by-step.