Square Root of 2164

The square root is the inverse of the square of the number. 2164 is not a perfect square. The square root of 2164 is expressed in both radical and exponential form. In the radical form, it is expressed as √2164, whereas (2164)^(1/2) in the exponential form. √2164 ≈ 46.517, 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 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 2164 is broken down into its prime factors:

Step 1: Finding the prime factors of 2164 Breaking it down, we get 2 x 2 x 541: 2^2 x 541

Step 2: Now we have found the prime factors of 2164. The second step is to make pairs of those prime factors. Since 2164 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating √2164 using prime factorization is impossible.

The long division method is particularly used for non-perfect square numbers. In this method, we 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 2164, we need to group it as 64 and 21. Step 2: Now we need to find n whose square is less than or equal to 21. We can say n is ‘4’ because 4 x 4 = 16, which is less than 21. Now the quotient is 4, and after subtracting 16 from 21, the remainder is 5. Step 3: Now let us bring down 64, which is the new dividend. Add the old divisor with the same number 4 + 4 = 8, which will be our new divisor. Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 8n as the new divisor, and we need to find the value of n. Step 5: The next step is finding 8n * n ≤ 564. Let's consider n as 6, now 86 x 6 = 516. Step 6: Subtract 516 from 564, the difference is 48, and the quotient is 46. 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 4800. Step 8: Now we need to find the new divisor that is 930 because 930 x 5 = 4650. Step 9: Subtracting 4650 from 4800, we get the result 150. Step 10: Now the quotient is 46.5. Step 11: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero. So the square root of √2164 is approximately 46.52.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Let us learn how to find the square root of 2164 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √2164. The smallest perfect square less than 2164 is 2025, and the largest perfect square greater than 2164 is 2209. √2164 falls somewhere between 45 and 47.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Largest perfect square - smallest perfect square).

Applying the formula: (2164 - 2025) ÷ (2209 - 2025) = 139 ÷ 184 ≈ 0.76 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: 45 + 0.76 = 45.76.

Therefore, the approximate square root of 2164 is 46.52.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, which 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 a positive square root that has more prominence due to its practical uses. This is why it is known as the principal square root.

• Prime factorization: Prime factorization is the determination of the set of prime numbers which multiply together to give the original integer.

• Long division method: A method used to find the square roots of non-perfect squares by dividing the number and obtaining the quotient with decimal precision.