Square Root of 178

The square root is the inverse of the square of the number. 178 is not a perfect square. The square root of 178 is expressed in both radical and exponential form. In radical form, it is expressed as √178, whereas (178)^(1/2) in exponential form. √178 ≈ 13.34166, 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 like 178, methods such as the long-division method and approximation method are used. Let us now learn these 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 178 is broken down into its prime factors:

Step 1: Finding the prime factors of 178 Breaking it down, we get 2 × 89: 2^1 × 89^1

Step 2: Now we have found the prime factors of 178. Since 178 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 178 using prime factorization alone is not feasible for finding an exact square root.

The long division method is particularly used for non-perfect square numbers. In this method, we find the square root using a step-by-step division process.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 178, we group it as 78 and 1.

Step 2: Now, find a number n whose square is less than or equal to 1. We use n = 1 because 1 × 1 ≤ 1. The quotient is 1, and after subtracting, the remainder is 0.

Step 3: Bring down 78 to form the new dividend. Add the old divisor with the same number: 1 + 1 = 2, which becomes the new divisor.

Step 4: The new divisor is 2n. We need to find a value for n such that 2n × n ≤ 78. Trying n = 3, we have 23 × 3 = 69.

Step 5: Subtracting 69 from 78 gives a remainder of 9. The quotient is 13.

Step 6: Since the dividend is less than the divisor, we add a decimal point and continue the process by bringing down pairs of zeros.

Step 7: Continue this process until we have enough decimal places. So the square root of √178 ≈ 13.341.

The approximation method is another way to find square roots. It is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 178 using the approximation method.

Step 1: Find the closest perfect squares surrounding 178. The closest perfect squares are 169 (13^2) and 196 (14^2). So √178 falls between 13 and 14.

Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). (178 - 169)/(196 - 169) = 9/27 ≈ 0.333 Using the formula, we identify the decimal point of our square root. The next step is adding the value we found initially to the decimal number: 13 + 0.333 ≈ 13.333, so the square root of 178 is approximately 13.333.

• Square root: A square root is the inverse operation to squaring a number. For example, 4^2 = 16, and the inverse is the square root: √16 = 4.

• Irrational number: A number that cannot be expressed as a simple fraction, such as √178, is an irrational number.

• Principal square root: The principal square root is the non-negative square root of a number. For example, the principal square root of 178 is √178 ≈ 13.341.

• Prime factorization: Breaking down a number into its prime factors, such as 178 = 2 × 89.

• Long division method: A method to find the square root of non-perfect squares through a division-like process that involves multiple steps.