Square Root of 420

The square root is the inverse of the square of a number. 420 is not a perfect square. The square root of 420 can be expressed in both radical and exponential forms. In the radical form, it is expressed as √420, whereas in exponential form it is (420)^(1/2). √420 ≈ 20.4939, 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, 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 420 is broken down into its prime factors:

Step 1: Finding the prime factors of 420 Breaking it down, we get 2 × 2 × 3 × 5 × 7: 2^2 × 3^1 × 5^1 × 7^1

Step 2: Now we found out the prime factors of 420. The second step is to make pairs of those prime factors. Since 420 is not a perfect square, the digits of the number can’t be grouped in pairs completely.

Therefore, calculating the square root of 420 using prime factorization gives an approximation.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square numbers 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 420, we group it as 20 and 4.

Step 2: Find n whose square is less than or equal to 4. We can say n is 2 because 2 × 2 = 4. The quotient is 2, and after subtracting 4 - 4, the remainder is 0.

Step 3: Bring down 20, which becomes the new dividend. Add the old divisor with itself: 2 + 2 = 4, which will be our new divisor.

Step 4: We find 4n such that 4n × n ≤ 20. If n is 0, then 4 × 0 = 0.

Step 5: Subtract 0 from 20, yielding a remainder of 20.

Step 6: Add a decimal point to the quotient and bring down two zeros to make the new dividend 2000.

Step 7: The new divisor is 40n. Find n such that 40n × n is less than or equal to 2000. If n is 4, then 404 × 4 = 1616.

Step 8: Subtract 1616 from 2000, leaving a remainder of 384.

Step 9: Repeat the process until the desired decimal places are obtained.

The square root of 420 is approximately 20.49.

The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Let us learn how to find the square root of 420 using the approximation method.

Step 1: Find the closest perfect squares to √420. The closest perfect squares are 400 (20^2) and 441 (21^2). Thus, √420 falls between 20 and 21.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (420 - 400) / (441 - 400) = 20 / 41 ≈ 0.49. Add this decimal to the smaller whole number: 20 + 0.49 = 20.49.

Thus, the square root of 420 is approximately 20.49.

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

• Irrational number: An irrational number is a number that cannot be expressed as a fraction of two integers (p/q) where q is not zero.

• Radical form: A way of expressing numbers using the root symbol, such as √420.

• Perfect square: A number that is the square of an integer. For example, 16 is a perfect square because it is 4 squared (4 × 4).

• Approximation: The process of finding a value that is close to the exact value, often used with non-perfect squares.