Square Root of 293

The square root is the inverse of the square of the number. 293 is not a perfect square. The square root of 293 is expressed in both radical and exponential form. In the radical form, it is expressed as √293, whereas (293)^(1/2) in the exponential form. √293 ≈ 17.11724, 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 long division and approximation methods 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 293 is broken down into its prime factors.

Step 1: Finding the prime factors of 293 293 is a prime number itself. Thus, its prime factorization is simply 293.

Step 2: Since 293 is a prime number and not a perfect square, calculating its square root using prime factorization is not feasible.

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 293, we need to group it as 93 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 lesser than or equal to 2. Now the quotient is 1 after subtracting 2 - 1, the remainder is 1.

Step 3: Now let us bring down 93, which is the new dividend. Add the old divisor with the same number 1 + 1 we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 193. Let us consider n as 7; now 27 × 7 = 189.

Step 6: Subtract 193 from 189, the difference is 4, and the quotient is 17.

Step 7: 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 400.

Step 8: Now we need to find the new divisor, which is 34, because 341 × 1 = 341.

Step 9: Subtracting 341 from 400, we get the result 59.

Step 10: Now the quotient is 17.1.

Step 11: 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 √293 is approximately 17.11.

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 293 using the approximation method.

Step 1: Now we have to find the closest perfect square of √293. The smallest perfect square less than 293 is 289, and the largest perfect square greater than 293 is 324. √293 falls somewhere between 17 and 18.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (293 - 289) ÷ (324 - 289) = 4 / 35 ≈ 0.114. 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 17 + 0.114 ≈ 17.114. So the square root of 293 is approximately 17.114.

• Square root: A square root is the inverse of a square. Example: 4² = 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, where q is not equal to zero and p and q are integers.

• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Examples include 2, 3, 5, 7, 11, 13, etc.

• Approximation method: A method used to estimate the value of a square root by comparing it to nearby perfect squares and using interpolation.

• Long division method: A step-by-step technique to determine the square root of a number by dividing it into manageable parts, typically used for non-perfect squares.