Square Root of 1780

The square root is the inverse of the square of the number. 1780 is not a perfect square. The square root of 1780 is expressed in both radical and exponential form. In radical form, it is expressed as √1780, whereas (1780)^(1/2) in exponential form. √1780 ≈ 42.2029, 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 1780 is broken down into its prime factors.

Step 1: Finding the prime factors of 1780 Breaking it down, we get 2 x 2 x 5 x 89: 2^2 x 5^1 x 89^1

Step 2: Now we found the prime factors of 1780. The second step is to make pairs of those prime factors. Since 1780 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1780 using prime factorization is impossible.

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 1780, we need to group it as 80 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 x 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 80, which is the new dividend. Add the old divisor with the same number 4 + 4 to get 8, which will be part of our new divisor.

Step 4: We now need to find a digit x such that 8x multiplied by x is less than or equal to 180. We find x is 2 because 82 x 2 = 164, which is less than 180.

Step 5: Subtract 164 from 180, which gives 16. Bring down two zeroes to make it 1600. Now the quotient is 42.

Step 6: Repeat the process by adding a decimal point to the quotient, and continue with the long division steps to find the decimal value to the desired accuracy.

We find the square root of 1780 is approximately 42.2029.

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 1780 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √1780. The smallest perfect square less than 1780 is 1764 (42^2), and the largest perfect square more than 1780 is 1849 (43^2). √1780 falls somewhere between 42 and 43.

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

Applying the formula (1780 - 1764) / (1849 - 1764) = 16 / 85 ≈ 0.1882 Using the formula, we identify the decimal point of our square root.

The next step is adding the value we got initially to the decimal number, which is 42 + 0.1882 ≈ 42.2029, so the square root of 1780 is approximately 42.2029.

• Square root: A square root is the inverse of a square. For 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 usually the positive square root that has more prominence due to its uses in the real world. This is known as the principal square root.

• Prime factorization: The process of breaking down a composite number into its prime factors. For example, the prime factorization of 1780 is 2^2 x 5^1 x 89^1.

• Long division method: A mathematical method used to find the square root of a non-perfect square number by dividing it into groups and finding the quotient and remainder step by step.