Square Root of 479

The square root is the inverse of the square of the number. 479 is not a perfect square. The square root of 479 is expressed in both radical and exponential form.

In the radical form, it is expressed as √479, whereas in exponential form as (479)^(1/2). √479 ≈ 21.877, 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 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 479 is broken down into its prime factors.

Step 1: Finding the prime factors of 479 The number 479 is a prime number, so it cannot be broken down further into prime factors.

Step 2: Since 479 is not a perfect square, calculating the square root using prime factorization directly 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 479, we need to group it as 79 and 4.

Step 2: Now we need to find n whose square is less than or equal to 4. We can say n is '2' because 2 x 2 = 4. Now the quotient is 2, and the remainder is 0.

Step 3: Now let us bring down 79, which is the new dividend. Add the old divisor with the same number 2 + 2 to get 4 as our new divisor.

Step 4: The new divisor is 4n. We need to find the value of n such that 4n x n ≤ 79. Let us consider n as 1. Now 41 x 1 = 41.

Step 5: Subtract 41 from 79, the difference is 38, 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 3800.

Step 7: Now we need to find the new divisor. We take 421 and find n such that 421n x n ≤ 3800. Let us consider n as 9. Now 421 x 9 = 3789.

Step 8: Subtracting 3789 from 3800, we get 11.

Step 9: Continue doing these steps until we get two numbers after the decimal point.

So the square root of √479 is approximately 21.87.

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 479 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √479. The smallest perfect square less than 479 is 441 (since 21² = 441) and the largest perfect square greater than 479 is 484 (since 22² = 484). √479 falls somewhere between 21 and 22.

Step 2: Now we need to apply the approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Using the formula: (479 - 441) / (484 - 441) = 38 / 43 ≈ 0.8837 Adding this to the smaller integer root, we get 21 + 0.8837 ≈ 21.8837.

So the square root of 479 is approximately 21.8837.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse operation is the square root, √16 = 4.
   
• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Example: 479.
   
• Irrational number: An irrational number cannot be written in the form of p/q, where q is not equal to zero, and p and q are integers.
   
• Approximation: Approximation involves finding a value that is close enough to the right answer, usually with some thought involved to get that value.
   
• Perfect square: A perfect square is an integer that is the square of an integer. Example: 441 is a perfect square because it is 21².