Square Root of 9216

The square root is the inverse of the square of the number. 9216 is a perfect square. The square root of 9216 is expressed in both radical and exponential forms. In radical form, it is expressed as √9216, whereas (9216)^(1/2) in exponential form. √9216 = 96, 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 can be used for perfect square numbers like 9216. When a number is a perfect square, the prime factorization method helps identify its square root. 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 9216 is broken down into its prime factors.

Step 1: Finding the prime factors of 9216

Breaking it down, we get 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 = 2^6 × 3^4

Step 2: Pair the prime factors. Since 9216 is a perfect square, we can pair all the prime factors.

Step 3: Multiply the factors in one group from each pair: (2^3 × 3^2 = 8 × 9 = 72). Since 9216 is a perfect square, the square root of 9216 is 96.

The long division method is particularly useful 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 9216, it is grouped as 92 and 16.

Step 2: Now we need to find a number whose square is less than or equal to the leftmost group, 92. We choose 9 since 9 × 9 = 81 ≤ 92. The quotient is 9, and the remainder is 11 after subtracting 81 from 92.

Step 3: Bring down the next pair, 16, to make the new dividend 1116. Add the old divisor with the same number 9 + 9, we get 18 which will be our new divisor.

Step 4: The new divisor will be 18n. We need to find n such that 18n × n ≤ 1116. Let n = 6, then 186 × 6 = 1116.

Step 5: Subtract 1116 from 1116; the remainder is 0.

The quotient is 96. Since the remainder is 0, the square root of 9216 is 96.

The approximation method is another way to find the square roots, and it is an easy method to find the square root of a given number. Since 9216 is a perfect square, the approximation method confirms the result but is not typically necessary for perfect squares.

Step 1: Identify the closest perfect square numbers around 9216. In this case, 9216 is itself a perfect square.

Step 2: Since we already found that √9216 is 96 using other methods, the approximation is not needed for this perfect square.

Thus, the square root of 9216 is 96.

• Square root: A square root is the inverse of a square. For example, 10^2 = 100, and the inverse of the square is the square root, which is √100 = 10.
   
• Perfect square: A perfect square is a number that can be expressed as the square of an integer. For example, 9216 is a perfect square because it can be expressed as 96^2.
   
• Rational number: A rational number is a number that can be written in the form of p/q, where q is not equal to zero, and p and q are integers.
   
• Long division method: This is a method used to find the square root of a number by dividing the number into groups and calculating the quotient systematically.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors, which can help in determining the square root of perfect squares.