Square Root of 17.8

The square root is the inverse operation of squaring a number. 17.8 is not a perfect square. The square root of 17.8 can be expressed in both radical and exponential forms. In radical form, it is expressed as √17.8, whereas in exponential form it is (17.8)^(1/2). The square root of 17.8 is approximately 4.21637, 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.

The prime factorization method is suitable for perfect squares. However, for non-perfect squares like 17.8, methods such as the long-division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

Prime factorization involves expressing a number as a product of prime numbers. Since 17.8 is not an integer, prime factorization is not applicable directly. Therefore, calculating the square root of 17.8 using prime factorization is not possible.

The long division method is useful for non-perfect square numbers. This method allows us to find the square root more accurately. Here's how to find the square root using the long division method, step by step:

Step 1: Begin by grouping the digits of the number from right to left. For 17.8, we consider it as 1780 (after multiplying by 100 to remove the decimal).

Step 2: Find a number whose square is less than or equal to 17. The number is 4 because 4^2 = 16.

Step 3: Subtract 16 from 17, leaving a remainder of 1. Bring down the next pair, 80, to make it 180.

Step 4: Double the quotient (4), which gives us 8, and use it as the first digit of our new divisor. Find a number n such that 8n × n is less than or equal to 180.

Step 5: The number n is 2 because 82 × 2 = 164, and subtracting 164 from 180 gives a remainder of 16.

Step 6: Add a decimal point to the quotient and bring down 00, making it 1600.

Step 7: The next divisor is 84n, find n such that 84n × n is less than or equal to 1600.

Step 8: Continue this process until you achieve the desired precision.

The square root of 17.8 is approximately 4.21637.

The approximation method is a straightforward way to estimate the square root of a given number.

Step 1: Identify two perfect squares between which 17.8 falls.

The closest perfect squares are 16 and 25.

Thus, √17.8 is between 4 and 5.

Step 2: Apply the approximation formula:

(Given number - smaller perfect square) / (Larger perfect square - smaller perfect square).

Using this formula: (17.8 - 16) / (25 - 16) = 1.8 / 9 = 0.2

Step 3: Add the smaller square root (4) to the decimal value obtained: 4 + 0.2 = 4.2.

So, the approximate square root of 17.8 is 4.2.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, √16 = 4.
   
• Irrational number: An irrational number cannot be represented as a simple fraction. It has non-repeating, non-terminating decimal expansions. Example: √17.8 is irrational.
   
• Decimal: A decimal is a number that has a whole number part and a fractional part separated by a decimal point, such as 4.21637.
   
• Long division method: A method used to find square roots of non-perfect squares by systematically dividing and averaging.
   
• Perfect square: A perfect square is a number that is the square of an integer, like 16, which is 4^2.