Square Root of 158

The square root is the inverse of the square of the number. 158 is not a perfect square. The square root of 158 is expressed in both radical and exponential form. In the radical form, it is expressed as, √158, whereas (158)^(1/2) in the exponential form. √158 ≈ 12.5698, 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 158 is broken down into its prime factors.

Step 1: Finding the prime factors of 158 Breaking it down, we get 2 x 79: 2¹ x 79¹

Step 2: Now we found out the prime factors of 158. Since 158 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √158 using prime factorization alone is not practical.

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 158, we need to group it as 58 and 1.

Step 2: Now we need to find n whose square is less than or equal to 1. We use n = 1 because 1 x 1 is less than or equal to 1. Now the quotient is 1, and after subtracting 1 - 1, the remainder is 0.

Step 3: Now let us bring down 58, which is the new dividend. Add the old divisor with the same number, 1 + 1, to get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, and we need to find the value of n.

Step 5: Find 2n × n ≤ 58. Let us consider n as 2, now 2 x 2 x 2 = 48.

Step 6: Subtract 58 from 48; the difference is 10, and the quotient is 12.

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 1000.

Step 8: Now we need to find the new divisor that is 9 because 249 x 9 = 2241.

Step 9: Subtracting 2241 from 2400, we get the result 159.

Step 10: Now the quotient is 12.5.

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 √158 is approximately 12.57.

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 158 using the approximation method.

Step 1: Now we have to find the closest perfect square of √158. The smallest perfect square less than 158 is 144, and the largest perfect square greater than 158 is 169. √158 falls somewhere between 12 and 13.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula (158 - 144) ÷ (169 - 144) ≈ 0.56 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 12 + 0.56 = 12.56.

So the square root of 158 is approximately 12.57.

• 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, 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 is more prominent due to its uses in the real world. That is the reason it is also known as the principal square root.
   
• Factors: Factors are numbers you multiply together to get another number. For example, 2 and 79 are factors of 158.
   
• 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.