Square Root of 8354

The square root is the inverse of the square of a number. 8354 is not a perfect square. The square root of 8354 is expressed in both radical and exponential forms. In radical form, it is expressed as √8354, whereas (8354)^(1/2) is the exponential form. √8354 ≈ 91.421, 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, for non-perfect square numbers like 8354, 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 8354 is broken down into its prime factors.

Step 1: Finding the prime factors of 8354 Breaking it down, we get 2 x 3 x 1393.

Step 2: Now we found out the prime factors of 8354. Since 8354 is not a perfect square, the digits of the number can’t be grouped into pairs. Therefore, calculating 8354 using prime factorization is not straightforward.

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 8354, we need to group it as 54 and 83.

Step 2: Now we need to find n whose square is less than or equal to 83. We can say n as ‘9’ because 9 x 9 = 81, which is less than or equal to 83. Now the quotient is 9, and after subtracting, the remainder is 2.

Step 3: Bring down 54, which is the new dividend. Add the old divisor with the same number (9 + 9), we get 18, which will be part of our new divisor.

Step 4: Find 18n such that 18n x n ≤ 254. Let n be 1, so 181 x 1 = 181.

Step 5: Subtract 181 from 254; the remainder is 73.

Step 6: 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 7300.

Step 7: Continue this process to find more decimal places, eventually determining √8354 ≈ 91.421.

The approximation method is another way to find 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 8354 using the approximation method.

Step 1: We need to find the closest perfect squares to √8354. The closest perfect square less than 8354 is 8281 (91^2), and the closest greater perfect square is 8464 (92^2). √8354 falls somewhere between 91 and 92.

Step 2: Apply the formula

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (8354 - 8281) ÷ (8464 - 8281) = 0.421.

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 91 + 0.421 = 91.421, so the square root of 8354 is approximately 91.421.

• 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.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing the number into groups and using a systematic procedure to find the root.
   
• Prime factorization: Breaking down a number into its prime number factors.
   
• Approximation method: A method of estimating the square root of a non-perfect square by finding the nearest perfect squares and interpolating between them.