Square Root of 5314

The square root is the inverse of the square of the number. 5314 is not a perfect square. The square root of 5314 is expressed in both radical and exponential form. In the radical form, it is expressed as √5314, whereas (5314)^(1/2) in the exponential form. √5314 ≈ 72.9125, 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 5314 is broken down into its prime factors:

Step 1: Finding the prime factors of 5314 Breaking it down, we get 2 × 2657. 2657 is a prime number, so the complete factorization is 2 × 2657.

Step 2: Since 5314 is not a perfect square, the digits of the number cannot be grouped into pairs.

Therefore, calculating 5314 using prime factorization directly to find the square root is impractical.

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 5314, we need to group it as 53 and 14.

Step 2: Now we need to find n whose square is less than or equal to 53. We can say n is '7' because 7 × 7 = 49, which is the largest square less than 53. Now the quotient is 7, and the remainder is 53 - 49 = 4.

Step 3: Bring down the next pair of digits, 14, making the new dividend 414. Add the old divisor with itself, 7 + 7 = 14.

Step 4: Use 14 as the new divisor and find a digit n such that 14n × n is less than or equal to 414. Let's consider n as 2, then 142 × 2 = 284.

Step 5: Subtract 284 from 414, the difference is 130, and the quotient is 72.

Step 6: Since the dividend is still greater than the divisor, add a decimal point to the quotient and bring down two zeros to the remainder, making the new dividend 13000.

Step 7: Continue the process, finding the new divisor and subtracting, until the desired precision is reached.

Step 8: The approximate square root of √5314 is 72.9125.

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 5314 using the approximation method.

Step 1: Now we have to find the closest perfect square of √5314.

The smallest perfect square less than 5314 is 5184, and the closest perfect square greater than 5314 is 5329. √5314 falls somewhere between 72 and 73.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square)

Using the formula: (5314 - 5184) ÷ (5329 - 5184) = 130 ÷ 145 ≈ 0.8966 Adding this decimal to the integer part, we have 72 + 0.8966 ≈ 72.9125.

So, the square root of 5314 is approximately 72.9125.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, which 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 factorization: The expression of a number as a product of its prime factors.

• Decimal: If a number has a whole number and a fractional part in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.

• Long division method: A method used for finding square roots of numbers, especially non-perfect squares, by using division step by step to achieve a decimal value.