Square Root of 13154

The square root is the inverse of the square of a number. 13154 is not a perfect square. The square root of 13154 is expressed in both radical and exponential forms. In the radical form, it is expressed as √13154, whereas (13154)^(1/2) in the exponential form. √13154 ≈ 114.672, 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 13154 is broken down into its prime factors.

Step 1: Finding the prime factors of 13154 Breaking it down, we get 2 x 6577: 2¹ x 6577

Step 2: Now we found out the prime factors of 13154. Since 13154 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 13154 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 13154, we need to group it as 54 and 131.

Step 2: Now we need to find n whose square is less than or equal to 131. We can say n is ‘11’ because 11 x 11 = 121, which is less than 131. Now the quotient is 11, and after subtracting 121 from 131, the remainder is 10.

Step 3: Now let us bring down 54, which is the new dividend. Add 11 with itself to get 22, which will be our new divisor.

Step 4: The next step is finding n such that 22n x n is less than or equal to 1054.

Step 5: Let us consider n as 4, so 224 x 4 = 896.

Step 6: Subtract 896 from 1054, and the difference is 158, and the new quotient becomes 114.

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 15800.

Step 8: Now we need to find the new divisor. Add 4 to 224 to get 228, and find n such that 228n x n is less than or equal to 15800.

Step 9: Continue performing these steps until you get two digits after the decimal point. So the square root of √13154 is approximately 114.672.

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 13154 using the approximation method.

Step 1: Now we have to find the closest perfect square of √13154. The smallest perfect square less than 13154 is 12996 (114²) and the largest perfect square greater than 13154 is 13225 (115²). √13154 falls somewhere between 114 and 115.

Step 2: Now we need to apply linear interpolation between these two squares: Using the formula: (given number - smaller perfect square) / (larger perfect square - smaller perfect square)

(13154 - 12996) / (13225 - 12996) ≈ 0.672

Adding this decimal to the smaller integer root: 114 + 0.672 = 114.672

Thus, the square root of 13154 is approximately 114.672.

• 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, the positive square root is typically used more often due to its real-world applications. This is known as the principal square root.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.

• Long division method: A method to find the square root of a non-perfect square by breaking it down into smaller steps, involving division and subtraction.