Square Root of 183

The square root is the inverse of the square of the number. 183 is not a perfect square. The square root of 183 is expressed in both radical and exponential form. In the radical form, it is expressed as √183, whereas in the exponential form, it is expressed as (183)^(1/2). √183 ≈ 13.5277, 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 183, 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 183 is broken down into its prime factors.

Step 1: Finding the prime factors of 183 Breaking it down, we get 3 x 61, which are both prime numbers: 3^1 x 61^1

Step 2: Now we have found the prime factors of 183. Since 183 is not a perfect square, these factors cannot form a pair.

Therefore, calculating 183 using prime factorization is not feasible for finding an exact square root.

The long division method is used for non-perfect square numbers. In this method, we should find the closest perfect square number to the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: Begin by grouping the digits of 183 starting from the right. We have 83 and 1 as grouped numbers.

Step 2: Find a number whose square is less than or equal to 1. Here, 1^2 = 1 fits. Thus, the quotient is 1, and the remainder is 0 after subtracting 1 from 1.

Step 3: Bring down 83, making the new dividend 83. Add the old divisor and the quotient: 1 + 1 = 2, which becomes part of the new divisor.

Step 4: Determine the next number n such that 2n × n is less than or equal to 83. Trying n = 3 gives 23 × 3 = 69.

Step 5: Subtract 69 from 83, giving a remainder of 14. The quotient is now 13.

Step 6: Add a decimal point to the quotient and bring down 00 to the remainder, making it 1400.

Step 7: Find the new divisor: 26n, where n = 5, gives 265 × 5 = 1325.

Step 8: Subtract 1325 from 1400, yielding a remainder of 75. The quotient is now 13.5.

Step 9: Continue the process until you reach the desired accuracy.

The square root of 183 is approximately 13.53.

The approximation method is an easy method for finding square roots. Let us learn how to find the square root of 183 using this method.

Step 1: Identify the closest perfect squares around √183. The smallest perfect square less than 183 is 169, and the largest perfect square greater than 183 is 196. Therefore, √183 is between 13 and 14.

Step 2: Use interpolation to find a closer estimate. The formula is: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) Using the formula: (183 - 169) / (196 - 169) = 14 / 27 ≈ 0.5185

Step 3: Add this decimal to the smaller perfect square root: 13 + 0.5185 ≈ 13.52, so the square root of 183 is approximately 13.52.

• Square root: A square root is the inverse of squaring a number. For example, if 4² = 16, then the square root of 16 is 4.

• Irrational number: An irrational number cannot be precisely written as a simple fraction of two integers. For example, the square root of 183 is irrational.

• Long division method: A technique used to calculate square roots of non-perfect squares by dividing and iterating through steps to reach an approximate value.

• Approximation method: A method to estimate the square root by comparing it with nearby perfect squares and using interpolation for better accuracy.

• Prime factorization: Breaking down a number into its basic prime number components, useful for simplifying square roots.