Square Root of 188

The square root is the inverse operation of squaring a number. 188 is not a perfect square. The square root of 188 can be expressed in both radical and exponential forms. In radical form, it is expressed as √188, whereas in exponential form, it is written as (188)^(1/2). √188 ≈ 13.711, which is an irrational number because it cannot be expressed exactly as a fraction 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, methods like 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 188 can be broken down into its prime factors:

Step 1: Finding the prime factors of 188 Breaking it down, we get 2 × 2 × 47: 2^2 × 47^1

Step 2: Since 188 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating √188 using prime factorization is not straightforward.

The long division method is particularly useful for non-perfect square numbers. In this method, we 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, group the numbers from right to left. For 188, group it as 88 and 1.

Step 2: Find n whose square is less than or equal to 1. We choose n as 1 because 1 × 1 = 1. Now the quotient is 1 and after subtracting 1-1, the remainder is 0.

Step 3: Bring down 88, making it the new dividend. Add the old divisor with itself: 1 + 1 = 2, which becomes the new divisor.

Step 4: The new divisor will be 2n. Find n such that 2n × n ≤ 88. Let n be 4, then 24 × 4 = 96.

Step 5: Subtract 88 from 96, the difference is -8, which indicates 96 is too large. Therefore, adjust n to 3, as 23 × 3 = 69.

Step 6: Subtract 69 from 88, the difference is 19, and the quotient is 13.

Step 7: As the dividend is less than the divisor, add a decimal point and bring down pairs of zero. The new dividend is 1900.

Step 8: Find the new divisor, which is 263. Choose n as 7 because 2637 × 7 = 18459.

Step 9: Subtract 18459 from 19000, the remainder is 541.

Step 10: Continue these steps until you get the desired decimal places. The square root of 188 is approximately 13.711.

The approximation method is another way to find square roots. It is an easy method to estimate the square root of a given number. Let us learn how to find the square root of 188 using this method:

Step 1: Identify the closest perfect squares to √188. The closest perfect square less than 188 is 169, and greater is 196. √188 falls between 13 and 14.

Step 2: Apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Using the formula: (188 - 169) / (196 - 169) = 19 / 27 ≈ 0.704 Adding this to the smaller perfect square's root: 13 + 0.704 = 13.704, so the square root of 188 is approximately 13.704.

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

• Irrational number: An irrational number cannot be written as a simple fraction of two integers. Example: √2, √3.

• Principal square root: A number has both positive and negative square roots, but the positive square root is most commonly used in real-world applications, known as the principal square root.

• Decimals: Numbers that contain a whole number and a fractional part separated by a decimal point. Examples include 7.86, 8.65, 9.42.

• Non-perfect square: A number that does not have an integer as its square root. Example: 188 is a non-perfect square as √188 is not an integer.