Square Root of 486

The square root is the inverse of the square of a number. 486 is not a perfect square. The square root of 486 is expressed in both radical and exponential form.

In radical form, it is expressed as √486, whereas in exponential form it is expressed as (486)^(1/2). The square root of 486 is approximately 22.045, 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 486 is broken down into its prime factors.

Step 1: Finding the prime factors of 486

Breaking it down, we get 2 x 3 x 3 x 3 x 3 x 3: 2 x 3⁵

Step 2: Now we found out the prime factors of 486. The second step is to make pairs of those prime factors. Since 486 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating the square root of 486 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 486, we group it as 86 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 after subtracting 4 - 4, the remainder is 0.

Step 3: Now let us bring down 86, which is 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. We need to find the value of n such that 4n x n ≤ 86. Let us consider n as 2, now 42 x 2 = 84.

Step 5: Subtract 86 from 84; the difference is 2, and the quotient is 22.

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 200.

Step 7: Now we need to find the new divisor that is 441 because 441 x 4 = 1764.

Step 8: Subtracting 1764 from 2000, we get the result 236.

Step 9: Now the quotient is 22.4.

Step 10: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √486 is approximately 22.04.

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 486 using the approximation method.

Step 1: Now, we have to find the closest perfect squares for √486. The smallest perfect square less than 486 is 484, and the largest perfect square greater than 486 is 529. √486 falls somewhere between 22 and 23.

Step 2: Now we need to apply the interpolation formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). Going by the formula, (486 - 484) ÷ (529 - 484) = 2/45 ≈ 0.044.

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 22 + 0.044 = 22.044, so the square root of 486 is approximately 22.044.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, which 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.
   
• Approximation: The process of finding a value that is close enough to the right answer, usually within an accepted range.
   
• Perfect square: A perfect square is a number that is the square of an integer. Example: 4, 9, 16 are perfect squares.
   
• Exponential form: A way of expressing a number using a base and an exponent. For example, the exponential form of 486 is 486^(1/2) when expressed as a square root.