Square Root of 13284

The square root is the inverse of the square of the number. 13284 is not a perfect square. The square root of 13284 is expressed in both radical and exponential form. In the radical form, it is expressed as √13284, whereas (13284)^(1/2) in the exponential form. The square root of 13284 is approximately 115.248, 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 the long-division method and approximation method are used. Let us now learn the following methods:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 13284 is broken down into its prime factors:

Step 1: Finding the prime factors of 13284 Breaking it down, we get 2 x 2 x 3 x 3 x 3 x 7 x 11.

Step 2: Now we found out the prime factors of 13284. The second step is to make pairs of those prime factors. Since 13284 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 13284 using prime factorization is impossible.

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 13284, we need to group it as 84 and 132.

Step 2: Now we need to find n whose square is closest to 132. We can say n as ‘11’ because 11 x 11 is 121, which is less than or equal to 132. Now the quotient is 11, and after subtracting 121 from 132, the remainder is 11.

Step 3: Now let us bring down 84 to make it the new dividend, which becomes 1184. Add the old divisor with the same number 11 + 11 to get 22, which will be our new divisor.

Step 4: The new divisor will be the sum of the previous divisor and the quotient. Now we get 22n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 22n x n ≤ 1184. Let us consider n as 5, now 225 x 5 = 1125.

Step 6: Subtracting 1125 from 1184 gives us a difference of 59, and the quotient is 115.

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. The new dividend becomes 5900.

Step 8: Now we need to find the new divisor, which is 1154, because 1154 x 4 = 4616.

Step 9: Subtracting 4616 from 5900 gives a result of 1284.

Step 10: Now the quotient is 115.2.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue until the remainder is zero. So the square root of √13284 is approximately 115.25.

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 13284 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √13284. The smallest perfect square less than 13284 is 12996 (114²), and the largest perfect square greater than 13284 is 13689 (117²). √13284 falls somewhere between 114 and 117.

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

Using the formula (13284 - 12996) ÷ (13689 - 12996) = 288 ÷ 693 ≈ 0.415 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 114 + 0.415 = 114.415, so the square root of 13284 is approximately 115.25.

• Square root: A square root is the inverse of squaring a number. Example: 4² = 16 and the inverse of squaring is the square root, so √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, the positive square root is more commonly used in the real world. It is known as the principal square root.

• Prime factorization: It is the method of expressing a number as the product of its prime factors.

• Decimal: A decimal is a number that has a whole number part and a fractional part, separated by a decimal point, such as 7.86, 8.65, and 9.42.