Square Root of 445

The square root is the inverse of the square of the number. 445 is not a perfect square. The square root of 445 is expressed in both radical and exponential form. In radical form, it is expressed as √445, whereas (445)^(1/2) is the exponential form. √445 ≈ 21.095, 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, methods like long division and approximation 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 445 is broken down into its prime factors.

Step 1: Finding the prime factors of 445 Breaking it down, we get 5 x 89: 5^1 x 89^1

Step 2: Now we found the prime factors of 445. Since 445 is not a perfect square, it cannot be grouped into pairs.

Therefore, calculating √445 using prime factorization is not straightforward.

The long division method is 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 the case of 445, it is grouped as 45 and 4.

Step 2: Find n whose square is less than or equal to 4. We can say n as ‘2’ because 2 x 2 = 4. Now the quotient is 2; after subtracting 4 from 4, the remainder is 0.

Step 3: Bring down 45, making the new dividend 45. Add the old divisor with the same number 2 + 2 = 4, which will be our new divisor.

Step 4: Find 4n such that 4n x n is less than or equal to 45. Consider n as 1, then 41 x 1 = 41.

Step 5: Subtract 41 from 45, the difference is 4, and the quotient is 21.

Step 6: Since the dividend is less than the divisor, add a decimal point, allowing two zeroes to be added to the dividend. The new dividend is 400.

Step 7: Find the new divisor which completes the equation. Continue the long division process until you get sufficient decimal places.

So the square root of √445 ≈ 21.095.

The approximation method is another way to find square roots. It is an easy method to estimate the square root of a given number. Let's learn how to find the square root of 445 using the approximation method.

Step 1: Find the closest perfect squares around 445. The smallest perfect square less than 445 is 441, and the largest perfect square greater than 445 is 484. √445 falls between 21 and 22.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula (445 - 441) / (484 - 441) = 4 / 43 = 0.093 Adding this to the base value, 21 + 0.093 = 21.093.

So, the square root of 445 is approximately 21.093.

• Square root: A square root is the inverse of squaring a number. Example: 4² = 16, and the inverse is the square root, √16 = 4.

• Irrational number: An irrational number cannot be written in the form p/q, where q is not zero and p and q are integers.

• Principal square root: A number has both positive and negative square roots, but the positive square root is often used in practical applications. This is known as the principal square root.

• Prime factorization: Prime factorization is breaking down a number into its simplest prime factors. For example, the prime factorization of 445 is 5 x 89.

• Long division method: A method used to find the square root of non-perfect squares by dividing the number into groups and iterating through division and subtraction steps.