Square Root of 414

The square root is the inverse of squaring a number. 414 is not a perfect square. The square root of 414 can be expressed in both radical and exponential form. In radical form, it is expressed as √414, whereas in exponential form it is expressed as (414)^(1/2). √414 ≈ 20.346, which is an irrational number because it cannot be expressed as a fraction of two integers.

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, the long division method and approximation method are typically 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 414 is broken down into its prime factors:

Step 1: Finding the prime factors of 414 Breaking it down, we get 2 × 3 × 3 × 23: 2^1 × 3^2 × 23^1

Step 2: Now we have found the prime factors of 414. The second step is to make pairs of those prime factors. Since 414 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating the square root of 414 using prime factorization alone is not feasible.

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 414, we need to group it as 14 and 4.

Step 2: Now we need to find n whose square is 4. We can say n is ‘2’ because 2 × 2 is lesser than or equal to 4. Now the quotient is 2, and after subtracting 4-4, the remainder is 0.

Step 3: Now let us bring down 14, which is the new dividend. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

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

Step 5: The next step is finding 4n × n ≤ 14. Let us consider n as 3. Now 4 × 3 × 3 = 36, which exceeds 14, so we consider n as 2.

Step 6: Subtract 14 from 8 (4 × 2 × 2), and the difference is 6. The quotient is 20.

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 zeros to the dividend. Now the new dividend is 600.

Step 8: Now we need to find the new divisor that is close to 600. The best fit is 204 because 204 × 2 = 408. Step 9: Subtracting 408 from 600, we get the result 192.

Step 10: Now the quotient is 20.3.

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

So the square root of √414 is approximately 20.346.

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 414 using the approximation method.

Step 1: Now we have to find the closest perfect square of √414. The smallest perfect square less than 414 is 400, and the largest perfect square greater than 414 is 441. √414 falls somewhere between 20 and 21.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square) Using the formula (414 - 400) ÷ (441 - 400) = 0.34 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 20 + 0.34 = 20.34.

Therefore, the square root of 414 is approximately 20.34.

• Square root: A square root is the inverse operation of squaring a number. For example: 4^2 = 16, and the inverse operation is the square root, √16 = 4.

• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction, where the denominator is not zero and both numerator and denominator are integers.

• Principal square root: A number has both positive and negative square roots, but the positive square root is typically used in real-world applications, known as the principal square root.

• Decimal: A decimal number has a whole number part and a fractional part separated by a decimal point. For example: 7.86, 8.65, and 9.42 are decimals.

• Long division method: A method used to divide large numbers and also to find square roots of non-perfect squares through systematic division and subtraction steps.