Square Root of 291

The square root is the inverse of the square of the number. 291 is not a perfect square. The square root of 291 is expressed in both radical and exponential form. In radical form, it is expressed as √291, whereas in exponential form it is expressed as (291)^(1/2). √291 ≈ 17.05872, 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 typically used for perfect square numbers. However, for non-perfect square numbers like 291, the long division method and approximation method are used. Let us now explore 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 291 is broken down into its prime factors.

Step 1: Finding the prime factors of 291.

Breaking it down, we get 3 x 97: 3¹ x 97¹.

Step 2: Now we found out the prime factors of 291. Since 291 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 291 using prime factorization alone is not straightforward.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square numbers around 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 291, we group it as 91 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 2. Now the quotient is 1 and after subtracting, the remainder is 1.

Step 3: Bring down 91, making the new dividend 191. Add the old divisor with the same number: 1 + 1 = 2, which will be our new divisor.

Step 4: Find a digit for n such that 2n × n ≤ 191. Let's consider n as 7, now 27 × 7 = 189.

Step 5: Subtract 189 from 191, the remainder is 2, and the quotient is 17.

Step 6: As the dividend is less than the divisor, we add a decimal point and bring down two zeros, making it 200.

Step 7: Find a new digit for n using 340 as the new divisor. Continuing these steps until two decimal places, we find that the square root of 291 is approximately 17.058.

The approximation method is another method for finding square roots; it is a simple way to estimate the square root of a given number. Let's learn how to find the square root of 291 using the approximation method.

Step 1: Find the closest perfect squares to 291. The smallest perfect square less than 291 is 289, and the largest perfect square greater than 291 is 324. √291 falls somewhere between 17 and 18.

Step 2: Apply the formula (Given number - smallest perfect square) / (Larger perfect square - smallest perfect square). Using the formula (291 - 289) / (324 - 289) = 0.057. Adding this to 17 (the square root of 289), we get 17 + 0.057 = 17.057. Thus, the approximate square root of 291 is 17.057.

• Square root: A square root is the inverse of squaring a number. For example, 4² = 16, and the inverse is the square root, so √16 = 4.

• Irrational number: An irrational number is a number that cannot be written as a simple fraction (p/q). An example is √2.

• Principal square root: The principal square root is the non-negative square root of a number. It is the most commonly used square root in practical applications.

• Prime factorization: Prime factorization involves expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 × 3².

• Decimal approximation: A decimal approximation provides a non-exact, rounded value for irrational numbers, useful for practical calculations. For instance, √291 ≈ 17.05872.