Square Root of 8192

The square root is the inverse of the square of a number. 8192 is not a perfect square. The square root of 8192 is expressed in both radical and exponential form. In the radical form, it is expressed as √8192, whereas (8192)^(1/2) in the exponential form. √8192 ≈ 90.5097, 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 8192 is broken down into its prime factors.

Step 1: Finding the prime factors of 8192 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2: 2^13

Step 2: Now we found the prime factors of 8192. The second step is to make pairs of those prime factors. Since 8192 is not a perfect square, the digits of the number can’t be grouped in equal pairs completely. Therefore, calculating 8192 using prime factorization is not directly possible.

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 8192, we need to group it as 92 and 81.

Step 2: Now we need to find n whose square is closest to 81. We can say n as ‘9’ because 9 x 9 = 81 is lesser than or equal to 81. Now the quotient is 9, and after subtracting 81-81 the remainder is 0.

Step 3: Now let us bring down 92, which is the new dividend. Add the old divisor with the same number, 9 + 9 = 18, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 18n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 18n x n ≤ 9200. Let us consider n as 5, now 185 x 5 = 925.

Step 6: Subtract 9200 from 925, the difference is 8275, and the new quotient is 90.5.

Step 7: 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 827500.

Step 8: Now we need to find the new divisor that is 180 because 18005 x 5 = 90025.

Step 9: Subtracting 90025 from 827500, we get the result 737475.

Step 10: Now the quotient is 90.50.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue until the remainder is zero. So the square root of √8192 is approximately 90.51.

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 8192 using the approximation method.

Step 1: Now we have to find the closest perfect square of √8192. The smallest perfect square less than 8192 is 8100, and the largest perfect square greater than 8192 is 8281. √8192 falls somewhere between 90 and 91.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Going by the formula (8192 - 8100) ÷ (8281 - 8100) = 92 / 181 ≈ 0.50.

Using the formula, we identified the decimal point of our square root.

The next step is adding the value we got initially to the decimal number, which is 90 + 0.50 = 90.50, so the square root of 8192 is approximately 90.50.

• 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, that 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.
   
• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
   
• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.