Square Root of 47

The square root is the inverse of the square of the number. 47 is not a perfect square. The square root of 47 is expressed in both radical and exponential form.

In the radical form, it is expressed as √47, whereas (47)^(1/2) in the exponential form. √47 ≈ 6.85565, 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, 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 47 is broken down into its prime factors:

Step 1: Finding the prime factors of 47 47 is a prime number, so it can only be divided by 1 and 47. Since 47 is not a perfect square, calculating √47 using prime factorization does not provide an exact integer result.

The long division method is particularly used for non-perfect square numbers. In this method, we check the closest perfect square numbers 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, group the numbers from right to left. For 47, we consider it as a whole number since it is less than 100.

Step 2: Find the largest number whose square is less than or equal to 47. We can say n as '6' because 6×6 = 36, which is less than 47. Now the quotient is 6, and after subtracting 36 from 47, the remainder is 11.

Step 3: Bring down a pair of zeroes, making the new dividend 1100. Add the old divisor with the same number (6+6) to get 12, which will be our new divisor.

Step 4: Find 'n' such that 12n × n ≤ 1100. In this case, n is 8, as 128×8 = 1024.

Step 5: Subtract 1024 from 1100, resulting in a remainder of 76. The quotient is now 6.8.

Step 6: Continue this process by bringing down more pairs of zeroes and finding the next digit. Following these steps, you continue until you achieve the desired level of accuracy.

The square root of 47 is approximately 6.855.

The approximation method is another method for finding the 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 47 using the approximation method.

Step 1: Find the closest perfect squares of √47. The closest perfect squares are 36 (6²) and 49 (7²). √47 falls somewhere between 6 and 7.

Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square)

Using the formula (47 - 36) ÷ (49 - 36) = 11/13 ≈ 0.846 Adding this to the smaller root: 6 + 0.846 = 6.846, so the square root of 47 is approximately 6.855.

• Square root: A square root of a number is one of the two equal factors of the number. For example, 4² = 16, and the inverse operation is the square root, which is √16 = 4.
   
• Irrational number: An irrational number cannot be written as a simple fraction (p/q) where q ≠ 0. For example, √47 is irrational.
   
• Prime number: A prime number is a number greater than 1 with no divisors other than 1 and itself. For example, 47 is a prime number.
   
• Decimal approximation: A method of expressing numbers by using a finite sequence of digits after the decimal point to approximate values like √47 ≈ 6.855.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 49 is a perfect square because it is 7².