Square Root of 338

The square root is the inverse of the square of the number. 338 is not a perfect square. The square root of 338 is expressed in both radical and exponential form. In the radical form, it is expressed as √338, whereas 338^(1/2) in the exponential form. √338 ≈ 18.3848, 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 338 is broken down into its prime factors.

Step 1: Finding the prime factors of 338 Breaking it down, we get 2 x 13 x 13: 2^1 x 13^2

Step 2: Now we found out the prime factors of 338. The second step is to make pairs of those prime factors. Since 338 is not a perfect square, therefore the digits of the number can’t be grouped into a perfect pair.

Therefore, calculating 338 using prime factorization is not straightforward, but we can simplify it to 13√2.

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 338, we need to group it as 38 and 3.

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

Step 3: Now let us bring down 38, 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 ≤ 238. Let us consider n as 8, now 28 x 8 = 224.

Step 6: Subtract 238 from 224; the difference is 14, and the quotient is 18.

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 1400.

Step 8: Now we need to find the new divisor. Let’s consider n as 4 because 368 x 4 = 1472.

Step 9: Subtracting 1472 from 1400, we get the result -72.

Step 10: Now the quotient is 18.3

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

So the square root of √338 is approximately 18.38.

The approximation method is another method for finding the 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 338 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √338.

The smallest perfect square less than 338 is 324, and the largest perfect square greater than 338 is 361. √338 falls somewhere between 18 and 19.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula: (338 - 324) ÷ (361 - 324) = 14 / 37 ≈ 0.378

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 18 + 0.38 = 18.38, so the square root of 338 is approximately 18.38.

• 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.

• Decimals: 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 expressing a number as the product of its prime numbers. For example, the prime factorization of 28 is 2 x 2 x 7.