Square Root of 1.8

The square root is the inverse of squaring a number. 1.8 is not a perfect square. The square root of 1.8 is expressed in both radical and exponential form. In radical form, it is expressed as √1.8, whereas in exponential form it is expressed as (1.8)^(1/2). √1.8 ≈ 1.34164, 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 1.8, methods such as the long-division method and approximation method are used. Let us now learn the following methods:

• Long division method
• Approximation method

The long division method is particularly used for non-perfect square numbers. Let us now learn how to find the square root using the long division method, step by step:

Step 1: Start with grouping the digits of 1.8. Since it has only one digit before the decimal, we treat it as 1.8.

Step 2: Now, find the largest number whose square is less than or equal to 1. The number is 1, as 1 × 1 = 1. Now, the quotient is 1, and after subtracting, the remainder is 0.

Step 3: Bring down 80 (since we add two zeros to the remainder after the decimal point). The new dividend is 80.

Step 4: Double the quotient and write it as the new divisor. So, 2 × 1 = 2.

Step 5: Find a digit n such that 2n × n is less than or equal to 80. Here, n = 3, since 23 × 3 = 69.

Step 6: Subtract 69 from 80. The remainder is 11.

Step 7: Bring down two zeros to make it 1100.

Step 8: The new divisor becomes 26. Find n such that 26n × n ≤ 1100. Here, n = 4, as 264 × 4 = 1056.

Step 9: Subtract 1056 from 1100. The remainder is 44. Step 10: Continue this process to get more decimal places.

The square root of 1.8 is approximately 1.34.

The approximation method is another technique for finding 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 1.8 using the approximation method.

Step 1: Find two perfect squares between which 1.8 falls. The closest perfect squares are 1 (1^2) and 4 (2^2). Therefore, √1.8 falls between 1 and 2.

Step 2: Apply the interpolation formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). For 1.8, the calculation is (1.8 - 1) / (4 - 1) = 0.8 / 3 ≈ 0.2667.

Step 3: Add this value to the smaller root: 1 + 0.2667 ≈ 1.267. Thus, the square root of 1.8 is approximately 1.34, when calculated more precisely.

• Square root: The square root is the inverse of squaring a number. For example, if 4^2 = 16, then √16 = 4.
   
• Irrational number: An irrational number is one that cannot be expressed as a simple fraction or ratio of two integers. For example, √2 is irrational.
   
• Approximation: Finding a value that is close to but not exactly equal to the true value. For example, √2 ≈ 1.414.
   
• Long division method: A process used to find the quotient of two numbers, often used to calculate square roots of non-perfect squares.
   
• Decimal: A number that has a fractional part separated by a decimal point, such as 1.34 or 3.56.