Square Root of 46656

The square root is the inverse of the square of the number. 46656 is a perfect square. The square root of 46656 is expressed in both radical and exponential form. In the radical form, it is expressed as √46656, whereas (46656)^(1/2) in the exponential form. √46656 = 216, which is a rational number because it can 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. For 46656, the prime factorization method is appropriate. 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 46656 is broken down into its prime factors.

Step 1: Finding the prime factors of 46656 Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3: 2^4 x 3^6

Step 2: Now we found out the prime factors of 46656. Since 46656 is a perfect square, we can pair the prime factors: (2^4 x 3^6) = (2^2 x 3^3)^2 = 216^2

Therefore, the square root of 46656 is 216.

The long division method is used for both perfect and 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 pairs. For 46656, we need to group it as 56, 66, and 4.

Step 2: Find a number whose square is less than or equal to the leftmost group of digits (4). The number is 2 because 2^2 = 4. Now the quotient is 2, and the remainder is 0.

Step 3: Bring down the next pair (66) to make the new dividend 66. Double the quotient (2) to get the new divisor part 4.

Step 4: Find a digit x such that 4x multiplied by x gives a number less than or equal to 66. The number is 1 because 41 x 1 = 41.

Step 5: Subtract 41 from 66, resulting in a remainder 25. Bring down the next pair (56) to make the new dividend 256.

Step 6: Double the quotient part obtained so far (21) to get 42. Find a digit x such that 42x multiplied by x gives a number less than or equal to 256. The number is 6 because 426 x 6 = 2556.

Step 7: Subtract 2556 from 256, resulting in a remainder 0.

Since there is no remainder and we have completed the division, the square root of 46656 is 216.

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 46656 using the approximation method.

Step 1: Identify perfect squares closest to 46656. The perfect square less than 46656 is 44100 (210^2), and the perfect square greater than 46656 is 48400 (220^2). √46656 falls somewhere between 210 and 220.

Step 2: However, since 46656 is a perfect square, we can use the midpoint of these two values to check, which is 216.

Step 3: Verify by squaring 216: 216 x 216 = 46656.

Therefore, the square root of 46656 is exactly 216.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16 and the inverse of the square is the square root, which is √16 = 4.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.
   
• Rational number: A rational number is a number that can be expressed in the form of p/q, where q is not equal to zero, and p and q are integers.
   
• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors. For example, the prime factorization of 36 is 2 x 2 x 3 x 3.
   
• Integer: An integer is a whole number that can be positive, negative, or zero. For example: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 are integers.