Square Root of 179

The square root is the inverse of the square of the number. 179 is not a perfect square. The square root of 179 is expressed in both radical and exponential form. In radical form, it is expressed as √179, whereas in exponential form, it is expressed as (179)^(1/2). √179 ≈ 13.3791, 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 179 is broken down into its prime factors:

Step 1: Finding the prime factors of 179 179 is a prime number itself, so its prime factorization is simply 179.

Step 2: Since 179 is not a perfect square, the prime factorization method cannot be used to easily calculate its square root.

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 179, we need to group it as 79 and 1.

Step 2: Now we need to find n whose square is 1. We can say n as ‘1’ because 1 × 1 is lesser than or equal to 1. Now the quotient is 1; after subtracting 1-1, the remainder is 0.

Step 3: Now let us bring down 79, which is the new dividend. Add the old divisor with the same number 1 + 1, we 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, we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 79. Let us consider n as 3, now 2 × 3 × 3 = 18.

Step 6: Subtract 79 from 18, the difference is 61, and the quotient is 13.

Step 7: Since the dividend is greater 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 6100.

Step 8: Now we need to find the new divisor that is 267 because 267 × 3 = 801.

Step 9: Subtracting 801 from 6100, we get the result 5299.

Step 10: Now the quotient is 13.3.

Step 11: 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 √179 is approximately 13.37.

The approximation method is another method for finding the 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 179 using the approximation method.

Step 1: Now we have to find the closest perfect square of √179. The smallest perfect square less than 179 is 169, and the largest perfect square greater than 179 is 196. √179 falls somewhere between 13 and 14.

Step 2: Now we need to apply the formula that is: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (179 - 169) ÷ (196 - 169) ≈ 0.37. 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 13 + 0.37 = 13.37, so the square root of 179 is approximately 13.37.

• 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.

• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

• Long division method: A method used to find the square root of non-perfect squares by a systematic division process.

• Decimal approximation: When a number is approximated using decimal places to represent the square root of a non-perfect square.