Square Root of 316

The square root is the inverse of the square of the number. 316 is not a perfect square. The square root of 316 is expressed in both radical and exponential form. In radical form, it is expressed as √316, whereas (316)^(1/2) is the exponential form. √316 ≈ 17.78279, 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 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 316 is broken down into its prime factors.

Step 1: Finding the prime factors of 316 Breaking it down, we get 2 × 2 × 79. Therefore, the prime factorization of 316 is 2^2 × 79^1.

Step 2: Now we found out the prime factors of 316. The second step is to make pairs of those prime factors. Since 316 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 316 using prime factorization 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 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: Group the numbers from right to left. In the case of 316, we need to group it as 16 and 3.

Step 2: Find n whose square is less than or equal to 3. We can say n is '1' because 1 × 1 is less than or equal to 3. The quotient is 1, and after subtracting 1 from 3, the remainder is 2.

Step 3: Bring down 16 to make the new dividend 216. Add the old divisor to the quotient, 1 + 1, to get 2, which becomes our new divisor.

Step 4: The new divisor will be the sum of the old divisor and a placeholder, making it 2n. We need to find n such that 2n × n ≤ 216. Let n be 7; now 27 × 7 = 189.

Step 5: Subtract 189 from 216, and the remainder is 27. The quotient is 17.

Step 6: Since the dividend is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend, making it 2700.

Step 7: The new divisor is 354 because 354 × 7 = 2478.

Step 8: Subtract 2478 from 2700 to get the remainder 222.

Step 9: Continue these steps until we get two numbers after the decimal point or until the remainder is zero.

The square root of √316 is approximately 17.78.

The approximation method is another way to find square roots. It is an easy method to find the square root of a given number. Let us learn how to find the square root of 316 using the approximation method.

Step 1: Find the closest perfect squares to √316.

The smallest perfect square less than 316 is 289, and the largest perfect square greater than 316 is 324. √316 falls somewhere between 17 and 18.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Using the formula, (316 - 289) / (324 - 289) = 27/35 ≈ 0.771. Adding this decimal to the integer part we initially estimated, we get 17 + 0.771 ≈ 17.771.

Thus, the square root of 316 is approximately 17.78.

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

• Irrational number: An irrational number cannot be expressed as a simple fraction, i.e., in the form p/q, where q ≠ 0 and p and q are integers.

• Principal square root: The principal square root is always the positive square root of a number, commonly used in calculations.

• Prime factorization: Breaking down a number into its prime factors. For example, the prime factorization of 316 is 2 × 2 × 79.

• Approximation: An approximate value is used when exact numbers are not possible, often used for irrational numbers like √316 ≈ 17.78.