Square Root of 4194

The square root is the inverse of the square of the number. 4194 is not a perfect square. The square root of 4194 is expressed in both radical and exponential form. In the radical form, it is expressed as √4194, whereas (4194)^(1/2) in the exponential form. √4194 ≈ 64.7542, 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 4194 is broken down into its prime factors:

Step 1: Finding the prime factors of 4194 Breaking it down, we get 2 x 3 x 11 x 11 x 7: 2^1 x 3^1 x 11^2 x 7^1

Step 2: Now we found out the prime factors of 4194. The second step is to make pairs of those prime factors. Since 4194 is not a perfect square, calculating 4194 using prime factorization involves taking the square root of the product of any pairs of factors.

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, group the numbers from right to left in pairs. In the case of 4194, we group it as 41 and 94.

Step 2: Now find a number whose square is less than or equal to 41. We can say '6' because 6 x 6 = 36, which is less than 41. Now the quotient is 6, and the remainder is 5.

Step 3: Bring down 94 to make the new dividend 594. Double the quotient and place it next to the divisor, making it 12.

Step 4: Find a digit 'n' such that 12n × n ≤ 594. Let's try '4'; 124 x 4 = 496.

Step 5: Subtract 496 from 594; the remainder is 98. The quotient is now 64.

Step 6: Since the dividend is less than the divisor, add a decimal point and two zeros. The new dividend is 9800.

Step 7: Continue with the long division to obtain the next digits after the decimal until the desired precision is reached.

So, the square root of √4194 ≈ 64.75.

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 4194 using the approximation method.

Step 1: Find the closest perfect squares to 4194. The smallest perfect square less than 4194 is 4096 (64^2), and the largest perfect square greater than 4194 is 4225 (65^2). So, √4194 is between 64 and 65.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). (4194 - 4096) ÷ (4225 - 4096) = 98 / 129 ≈ 0.76 Adding this to 64, we get 64 + 0.76 = 64.76, so the square root of 4194 is approximately 64.76.

• Square root: A square root is the inverse of a square. For 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 square numbers by dividing the number step by step.
   
• Prime factorization: Breaking down a number into its basic prime factors to simplify calculations or expressions.
   
• Approximation: A method used to estimate the value of a number when it cannot be calculated exactly, often used with irrational numbers like square roots.