Square Root of 6900

The square root is the inverse of the square of the number. 6900 is not a perfect square. The square root of 6900 is expressed in both radical and exponential form. In radical form, it is expressed as √6900, whereas 6900^(1/2) in exponential form. √6900 ≈ 83.0454, 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 6900 is broken down into its prime factors.

Step 1: Finding the prime factors of 6900 Breaking it down, we get 2 × 2 × 3 × 5 × 5 × 23: 2^2 × 3^1 × 5^2 × 23^1

Step 2: Now we found out the prime factors of 6900. The second step is to make pairs of those prime factors. Since 6900 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 6900 using prime factorization is not straightforward.

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 6900, we need to group it as 00 and 69.

Step 2: Now we need to find n whose square is less than or equal to 69. We can say n is ‘8’ because 8 × 8 = 64 is lesser than 69. Now the quotient is 8, and after subtracting 64 from 69, the remainder is 5.

Step 3: Now let us bring down 00 to make the new dividend 500. Add the old divisor with the same number 8 + 8 to get 16, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. We get 16n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 16n × n ≤ 500. Let us consider n as 3, now 16 × 3 × 3 = 489.

Step 6: Subtract 489 from 500, the difference is 11, and the quotient is 83.

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 1100.

Step 8: Now we need to find the new divisor. If we consider the new digit as 6, then 1666 × 6 = 9996.

Step 9: Subtracting 9996 from 11000, we get the result 1004.

Step 10: Now the quotient is 83.0

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 √6900 is approximately 83.045.

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 6900 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √6900. The smallest perfect square less than 6900 is 81^2 = 6561, and the largest perfect square greater than 6900 is 84^2 = 7056. √6900 falls somewhere between 81 and 84.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula, (6900 - 6561) ÷ (7056 - 6561) = 0.756 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: 81 + 0.756 = 81.756, so the square root of 6900 is approximately 83.045.

• Square root: A square root is the inverse of a square. Example: 4² = 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 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 a positive square root that has more prominence due to its uses in the real world. That is why it is also known as the principal square root.
   
• Prime factorization: Prime factorization is expressing a number as the product of prime numbers.
   
• Long division method: The long division method is a technique used to find the square root of non-perfect squares through repeated division.