Square Root of 1354

The square root is the inverse of the square of the number. 1354 is not a perfect square. The square root of 1354 is expressed in both radical and exponential form. In the radical form, it is expressed as √1354, whereas (1354)^(1/2) in the exponential form. √1354 ≈ 36.791, 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 long division and approximation methods 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 1354 is broken down into its prime factors:

Step 1: Finding the prime factors of 1354

Breaking it down, we get 2 x 677: 2^1 x 677^1

Step 2: Now we found out the prime factors of 1354. The second step is to make pairs of those prime factors. Since 1354 is not a perfect square, therefore, the digits of the number can’t be grouped in pairs. Therefore, calculating 1354 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 1354, we need to group it as 54 and 13.

Step 2: Now we need to find n whose square is less than or equal to 13. We can say n as ‘3’ because 3^2 = 9 is less than 13. Now the quotient is 3 after subtracting 9 from 13, the remainder is 4.

Step 3: Now let us bring down 54, which is the new dividend. Add the old divisor with the same number, 3 + 3, we get 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 454. Let us consider n as 7, now 67 × 7 = 469, which is greater than 454. So, we try n as 6, and 66 × 6 = 396, which is less than 454.

Step 6: Subtract 396 from 454, the difference is 58, and the quotient is 36.

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 5800.

Step 8: Now we need to find the new divisor that is 736 because 736 × 7 = 5152.

Step 9: Subtracting 5152 from 5800, we get the result 648.

Step 10: Now the quotient is 36.7.

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

So the square root of √1354 is approximately 36.79.

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 1354 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √1354. The smallest perfect square less than 1354 is 1296 and the largest perfect square greater than 1354 is 1369. √1354 falls somewhere between 36 and 37.

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 (1354 - 1296) ÷ (1369 - 1296) = 58 ÷ 73 ≈ 0.79. 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 36 + 0.79 = 36.79, so the square root of 1354 is approximately 36.79.

• Square root: A square root is the inverse of squaring a number. Example: 4^2 = 16, and the inverse of squaring 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 is used more frequently due to its applications in the real world. This is known as the principal square root.

• Prime factorization: Prime factorization is expressing a number as the product of its prime factors.

• Decimal: If a number has a whole number and a fractional part, it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.