Square Root of 263

The square root is the inverse of the square of the number. 263 is not a perfect square. The square root of 263 is expressed in both radical and exponential form. In the radical form, it is expressed as √263, whereas (263)^(1/2) in the exponential form. √263 ≈ 16.217, 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, the prime factorization method is not used for non-perfect square numbers where 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 263 is broken down into its prime factors. Step 1: Finding the prime factors of 263 263 is a prime number, so it is only divisible by 1 and itself. Therefore, calculating 263 using prime factorization is straightforward as it does not break down further.

The long division method is particularly used for non-perfect square numbers. In this method, we should 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, we need to group the numbers from right to left. In the case of 263, we need to group it as 63 and 2.

Step 2: Now we need to find n whose square is less than or equal to 2. We can say n as ‘1’ because 1 × 1 is less than or equal to 2. Now the quotient is 1, and after subtracting 1 from 2, the remainder is 1.

Step 3: Now let us bring down 63, making the new dividend 163. Add the old divisor with the same number 1 + 1, we get 2, which will be our new divisor.

Step 4: We need to find the new digit of the divisor such that 2n × n ≤ 163. Let us consider n as 6. Now 26 × 6 = 156.

Step 5: Subtract 156 from 163, the difference is 7, and the quotient is 16.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 700.

Step 7: Now we need to find the new divisor, which is 162. Multiply 162 × 4 = 648.

Step 8: Subtracting 648 from 700 we get the result 52.

Step 9: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue until the remainder is zero.

So the square root of √263 ≈ 16.217.

The approximation method is another method for finding square roots, it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 263 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √263. The smallest perfect square less than 263 is 256 and the largest perfect square greater than 263 is 289. √263 falls somewhere between 16 and 17.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Applying the formula (263 - 256) ÷ (289 - 256) = 7/33 ≈ 0.212 Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number which is 16 + 0.212 ≈ 16.217, so the square root of 263 is approximately 16.217.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16 and the inverse of the square is the square root, that is √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, q is not equal to zero and p and q are integers.
   
• Prime number: A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.
   
• Approximation: A method to find an estimated value that is close to the exact number but not exact.