Square Root of 480

The square root is the inverse of the square of the number. 480 is not a perfect square. The square root of 480 is expressed in both radical and exponential form.

In the radical form, it is expressed as √480, whereas (480)^(1/2) in the exponential form. √480 ≈ 21.9089, 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 and approximation methods 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 480 is broken down into its prime factors.

Step 1: Finding the prime factors of 480

Breaking it down, we get 2 × 2 × 2 × 2 × 2 × 3 × 5: 2⁵ × 3¹ × 5¹

Step 2: Now we found out the prime factors of 480. The second step is to make pairs of those prime factors. Since 480 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 480 using prime factorization is incomplete.

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 480, we need to group it as 80 and 4.

Step 2: Now we need to find n whose square is 4. We can say n as ‘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 80, which is the new dividend. Add the old divisor with the same number: 2 + 2, we get 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, we need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 80. Let us consider n as 5, now 4 × 5 × 5 = 100, which is too large. Consider n as 4, now 4 × 4 × 4 = 64.

Step 6: Subtract 80 from 64, the difference is 16, and the quotient is 24.

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 two zeroes to the dividend. Now the new dividend is 1600.

Step 8: Now we need to find the new divisor. We use 48 because 484 × 4 = 1936, which is too large. So we use 47 because 474 × 3 = 1422.

Step 9: Subtracting 1422 from 1600 we get the result 178.

Step 10: Now the quotient is 21.9

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

So the square root of √480 is approximately 21.91.

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 480 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √480. The smallest perfect square less than 480 is 441 and the largest perfect square greater than 480 is 484. √480 falls somewhere between 21 and 22.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula (480 - 441) ÷ (484 - 441) = 39 ÷ 43 ≈ 0.907

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 21 + 0.907 = 21.907, so the approximate square root of 480 is 21.907.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, that is, √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
   
• Prime factorization: The prime factorization of a number is the expression of the number as a product of its prime factors. Example: 12 = 2 × 2 × 3.
   
• Long division method: A method for finding the square root of a number by dividing and using a series of steps to find the square root to a desired degree of accuracy.