Square Root of 313

The square root is the inverse of the square of the number. 313 is not a perfect square. The square root of 313 is expressed in both radical and exponential form. In the radical form, it is expressed as √313, whereas in the exponential form it is expressed as (313)^(1/2). √313 ≈ 17.69181, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, 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 313 is broken down into its prime factors:

Step 1: Finding the prime factors of 313 313 is a prime number itself, so its prime factors are 313.

Step 2: Since 313 is not a perfect square, its digits cannot be grouped into pairs.

Therefore, calculating √313 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 313, we need to group it as 13 and 3.

Step 2: Now we need to find n whose square is less than or equal to 3. We can say n is ‘1’ because 1 × 1 is lesser than or equal to 3. Now the quotient is 1, after subtracting 1 from 3 the remainder is 2.

Step 3: Now let us bring down 13 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: Find a digit n such that 2n × n is less than or equal to 213. The suitable n is 8, giving us 28 × 8 = 224.

Step 5: Subtract 213 from 224, the difference is -11, and the quotient is 18.

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 1100.

Step 7: The new divisor becomes 36, and the divisor times the next digit should be less than or equal to 1100. Using 3, 363 × 3 = 1089.

Step 8: Subtracting 1089 from 1100, we get 11.

Step 9: Continue this process until you have the desired decimal places in the quotient.

The square root of √313 ≈ 17.691.

The approximation method is another way to find square roots, especially useful for quick estimates. Let's find the square root of 313 using this method.

Step 1: Find the closest perfect squares around 313.

The smallest perfect square below 313 is 289 (since 17^2 = 289), and the largest perfect square above 313 is 324 (since 18^2 = 324). Thus, √313 falls between 17 and 18.

Step 2: Apply the approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). (313 - 289) / (324 - 289) = 24 / 35 ≈ 0.686.

Step 3: Add this to the lower bound: 17 + 0.686 = 17.686, so √313 ≈ 17.686.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the square root is √16 = 4.

• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

• Irrational number: An irrational number cannot be written in the form of p/q, where q is not zero, and p and q are integers.

• Radical: The symbol (√) used to denote the square root or nth root of a number.

• Decimal: A number with a whole number and a fractional part separated by a decimal point, like 17.691.