Square Root of 161

A square root is the inverse operation of squaring a number. Since 161 is not a perfect square, its square root is expressed in both radical and exponential forms. In radical form, it is expressed as √161, whereas in exponential form it is expressed as (161)^(1/2). The square root of 161 is approximately 12.6886, which is an irrational number because it cannot be expressed as a simple fraction p/q, where p and q are integers and q ≠ 0.

For non-perfect square numbers, methods like the long division method and approximation method are used. Let's explore these methods:

• Long division method
• Approximation method

The long division method is used for finding the square roots of non-perfect square numbers. Here's how to find the square root of 161 using this method:

Step 1: Begin by setting up the number in pairs from right to left. For 161, it will be grouped as 61 and 1.

Step 2: Find a number n whose square is less than or equal to 1. The number n is 1 because 1^2 is less than or equal to 1. The quotient is 1, and the remainder is 0 after subtracting 1.

Step 3: Bring down 61 to form the new dividend. Add the previous divisor (1) to itself to get 2, which becomes the new divisor.

Step 4: Find n such that 2n × n ≤ 61. If n is 3, then 23 × 3 = 69, which is greater than 61. Try n = 2, then 22 × 2 = 44.

Step 5: Subtract 44 from 61 to get a remainder of 17. The quotient is now 12.

Step 6: Add a decimal point and bring down 00 to the remainder to make it 1700.

Step 7: The new divisor is 24 (old divisor 22 plus n), and this process continues to find more precise values.

The approximate square root of 161 using the long division method is 12.688.

The approximation method is another way to find square roots. Let's find the square root of 161 using this method:

Step 1: Identify the perfect squares closest to 161. The smallest is 144, and the largest is 169. Thus, √161 lies between 12 and 13.

Step 2: Apply the approximation formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Calculation: (161 - 144) / (169 - 144) = 17 / 25 = 0.68 Add this to the lower perfect square root: 12 + 0.68 = 12.68

So, the square root of 161 is approximately 12.68.

• Square root: The operation that finds a number which, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 x 5 = 25.
   
• Irrational number: A number that cannot be expressed as a simple fraction. Its decimal form is non-repeating and non-terminating, such as √161.
   
• Perfect square: A number that is the square of an integer. For example, 144 is a perfect square because it is 12².
   
• Radical: The symbol (√) used to denote the square root or nth root of a number.
   
• Approximation: A value or number that is close to but not exactly equal to the exact value, often used for irrational numbers like √161.