Square Root of 378

The square root is the inverse of the square of the number. 378 is not a perfect square. The square root of 378 is expressed in both radical and exponential form. In the radical form, it is expressed as √378, whereas (378)^(1/2) is in exponential form. √378 ≈ 19.4422, 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 378 is broken down into its prime factors.

Step 1: Finding the prime factors of 378 Breaking it down, we get 2 x 3 x 3 x 3 x 7: 2^1 x 3^3 x 7^1

Step 2: Now we found out the prime factors of 378.

The second step is to make pairs of those prime factors.

Since 378 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √378 using prime factorization requires approximation.

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 378, we need to group it as 78 and 3.

Step 2: Now, we need to find n whose square is 3. We can say n as ‘1’ because 1 x 1 is lesser than or equal to 3. Now the quotient is 1, and after subtracting 1 from 3, the remainder is 2.

Step 3: Now let us bring down 78, which is the new dividend. Add the old divisor with the same number, 1 + 1, to get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 2n x n ≤ 278. Let us consider n as 9, now 29 x 9 = 261.

Step 6: Subtract 261 from 278; the difference is 17, and the quotient is 19.

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 1700.

Step 8: Now we need to find that 389 x 4 is closest to 1700 without exceeding it.

Step 9: Subtracting 1556 from 1700, we get the result 144.

Step 10: Now the quotient is 19.4.

Step 11: Continue doing these steps until we get the desired precision.

Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √378 is approximately 19.44.

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 378 using the approximation method.

Step 1: Now we have to find the closest perfect square of √378. The smallest perfect square less than 378 is 361, and the largest perfect square greater than 378 is 400. √378 falls somewhere between 19 and 20.

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 (378 - 361) ÷ (400 - 361) = 17 ÷ 39 ≈ 0.44

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 19 + 0.44 = 19.44, so the square root of 378 is approximately 19.44.

• Square root: A square root is the inverse of squaring a number. For example, 4^2 = 16, and the inverse of the square is the square root, which 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.
   
• Prime factorization: Prime factorization is breaking down a number into its basic prime number factors, which multiply to the original number.
   
• Long division method: A systematic approach to dividing numbers to find the square root of non-perfect squares, involving grouping, estimating, and subtracting.
   
• Approximation method: A technique to estimate the square root of a number by finding its position between two perfect squares and refining the estimate to the desired precision.