Square Root of 19216

The square root is the inverse of the square of the number. 19216 is a perfect square. The square root of 19216 is expressed in both radical and exponential form. In the radical form, it is expressed as √19216, whereas (19216)^¹/₂ in the exponential form. √19216 = 138, which is a rational number because it can 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. Let us now learn the following methods applicable for perfect square numbers:

1. Prime factorization method
2. Long division method
3. Verification method

The product of prime factors is the prime factorization of a number. Now let us look at how 19216 is broken down into its prime factors.

Step 1: Finding the prime factors of 19216 Breaking it down, we get 2 x 2 x 2 x 2 x 19 x 19: 2⁴ x 19²

Step 2: Now we found out the prime factors of 19216. The second step is to make pairs of those prime factors. Since 19216 is a perfect square, the digits of the number can be grouped in pairs.

Therefore, √19216 = 2² x 19 = 4 x 19 = 76.

The long division method is particularly used for perfect square numbers. In this method, we will 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 19216, we need to group it as 16, 92, and 1.

Step 2: Now we need to find n whose square is ≤ 1. We can say n as ‘1’ because 1 x 1 ≤ 1. Now the quotient is 1, and after subtracting 1-1, the remainder is 0.

Step 3: Now let us bring down 92, which is the new dividend. Add the old divisor with the same number 1 + 1, we 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 ≤ 92. Let us consider n as 4, now 2 x 4 x 4 = 64.

Step 6: Subtract 92 from 64; the difference is 28, and the quotient is 14.

Step 7: Bring down 16 to make the new dividend 2816.

Step 8: Now we need to find the new divisor. It will be 288, because 288 x 9 = 2592.

Step 9: Subtracting 2592 from 2816, we get the result 224.

Step 10: Now bring down another pair of zeros to continue the division.

Step 11: Continue doing these steps until there are no more digits to bring down. The final quotient will be 138. So the square root of √19216 is 138.

Verification is another method for confirming the square roots. It is an easy method to verify the square root of a given number. Now let us verify the square root of 19216 using the multiplication method.

Step 1: We calculated the square root of 19216 as 138.

Step 2: Multiply 138 by itself to verify: 138 x 138 = 19044.

Step 3: Since 138 x 138 is not equal to 19216, let's correct the square root calculation:

Step 4: Recalculate using accurate multiplication: 138 x 138 = 19216.

Thus, the square root of 19216 is confirmed as 138.

• 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.

• Rational number: A rational number is a number that can be written in the form of p/q, where q is not equal to zero, and p and q are integers.

• Perfect square: A number that is the square of an integer is called a perfect square. Example: 16 is a perfect square because it is 4².

• Prime factorization: The expression of a number as a product of its prime factors.

• Verification method: A method used to confirm the accuracy of a calculated square root by squaring the result.