Square Root of 1768

The square root is the inverse of the square of the number. 1768 is not a perfect square. The square root of 1768 is expressed in both radical and exponential forms. In the radical form, it is expressed as √1768, whereas (1768)^(1/2) in the exponential form. √1768 ≈ 42.043, 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 1768 is broken down into its prime factors.

Step 1: Finding the prime factors of 1768 Breaking it down, we get 2 × 2 × 2 × 221: 2^3 × 221

Step 2: Now we found out the prime factors of 1768. The second step is to make pairs of those prime factors. Since 1768 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 1768 using prime factorization directly is not possible.

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 1768, we need to group it as 68 and 17.

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

Step 3: Now let us bring down 68 which is the new dividend. Add the old divisor with the same number 4 + 4 to get 8, which will be our new divisor's leading digit.

Step 4: The new divisor will be 8n. We need to find the value of n such that 8n × n ≤ 168. Let us consider n as 2, now 82 × 2 = 164.

Step 5: Subtract 164 from 168; the difference is 4, and the quotient is 42.

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 zeroes to the dividend. Now the new dividend is 400.

Step 7: Now we need to find the new divisor that is 841 + 2 = 842, and find n such that 842n × n ≤ 40000.

Step 8: Continue the process until you get two numbers after the decimal point.

So the square root of √1768 ≈ 42.043.

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 1768 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1768. The smallest perfect square less than 1768 is 1764 (42^2), and the largest perfect square greater than 1768 is 1849 (43^2). √1768 falls between 42 and 43.

Step 2: Now we need to apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). (1768 - 1764) ÷ (1849 - 1764) = 4 ÷ 85 ≈ 0.047. Using the formula, we identify the decimal point of our square root. The next step is adding the initial whole value to the decimal number, which is 42 + 0.047 ≈ 42.047.

So the square root of 1768 is approximately 42.047.

• 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 has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.

• Prime factorization: The process of breaking down a number into its basic building blocks, which are prime numbers. For example, the prime factorization of 1768 is 2 × 2 × 2 × 221.