Square Root of 446

The square root is the inverse of the square of a number. 446 is not a perfect square. The square root of 446 is expressed in both radical and exponential form. In the radical form, it is expressed as √446, whereas (446)^(1/2) in the exponential form. √446 ≈ 21.1187, 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 446, 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 446 is broken down into its prime factors.

Step 1: Finding the prime factors of 446 Breaking it down, we get 2 x 223, which means 446 = 2^1 x 223^1.

Step 2: Since 446 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 446 using prime factorization provides no integer result.

The long division method is particularly used for non-perfect square numbers. 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 446, we need to group it as 4 and 46.

Step 2: Find a number n whose square is ≤ 4. We can say n is 2 because 2 x 2 is less than or equal to 4. Now the quotient is 2, and after subtracting, the remainder is 0.

Step 3: Bring down the next group, which is 46, making it the new dividend. Add the old divisor with the same number, 2 + 2, to get 4, which will be our new divisor.

Step 4: The new divisor will be 4n. Find n such that 4n x n ≤ 46. Let us consider n as 1, then 41 x 1 = 41.

Step 5: Subtract 46 from 41, the difference is 5, and the quotient is 21.

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 500.

Step 7: Find the new divisor that is 423 because 421 x 1 = 421.

Step 8: Subtracting 421 from 500 gives 79.

Step 9: Continue doing these steps until we get two numbers after the decimal point. Suppose there are no decimal values, continue till the remainder is zero.

So the square root of √446 ≈ 21.12.

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 446 using the approximation method.

Step 1: Find the closest perfect squares to √446. The smallest perfect square less than 446 is 441, and the largest perfect square greater than 446 is 484. √446 falls somewhere between 21 and 22.

Step 2: Apply the formula (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Using the formula (446 - 441) ÷ (484 - 441) = 0.116 Adding the value we got initially to the decimal number, 21 + 0.116 = 21.116, so the square root of 446 is approximately 21.12.

• Square root: A square root is the inverse operation of squaring a number. For example, if 4² = 16, then √16 = 4.

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

• Long division method: A mathematical method used to find the square root of non-perfect squares through a systematic process of division.

• Approximation: A method of finding a value that is close to the actual value, often used for estimating square roots of non-perfect squares.

• Perfect square: A number that can be expressed as the square of an integer. For example, 16 is a perfect square because it is 4².