Square Root of 5040

The square root is the inverse of the square of a number. 5040 is not a perfect square. The square root of 5040 is expressed in both radical and exponential form. In the radical form, it is expressed as √5040, whereas (5040)^(1/2) in the exponential form. √5040 ≈ 70.933, 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 5040 is broken down into its prime factors.

Step 1: Finding the prime factors of 5040 Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 3 x 5 x 7: 2^4 x 3^2 x 5 x 7

Step 2: Now we found out the prime factors of 5040. The second step is to make pairs of those prime factors. Since 5040 is not a perfect square, the digits of the number can’t be grouped in pairs to form a square. Therefore, calculating √5040 using prime factorization directly is not 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 5040, we need to group it as 40 and 50.

Step 2: Now we need to find n whose square is less than or equal to 50. We can say n as ‘7’ because 7 x 7 = 49 is less than 50. Now the quotient is 7, and after subtracting 49 from 50, the remainder is 1.

Step 3: Now let us bring down 40, making the new dividend 140. Add the old divisor with the same number 7 + 7 = 14, which will be our new divisor.

Step 4: The new divisor will be 14n. Now we get 14n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 14n × n ≤ 140. Let us consider n as 9, now 14 x 9 = 126.

Step 6: Subtract 126 from 140; the difference is 14, and the quotient becomes 79.

Step 7: Since the remainder is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to bring down two zeroes to the dividend. Now the new dividend is 1400.

Step 8: Find the new divisor, which is 158 because 158 x 8 = 1264.

Step 9: Subtracting 1264 from 1400, we get the result 136.

Step 10: Now the quotient is 79.8.

Step 11: Continue doing these steps until we get the desired number of decimal places for accuracy. So the square root of √5040 is approximately 70.933.

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 5040 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √5040. The closest perfect squares are 4900 (70^2) and 5184 (72^2), so √5040 falls somewhere between 70 and 72.

Step 2: Now we need to apply the formula for linear approximation: (Given number - smallest perfect square) ÷ (Largest perfect square - smallest perfect square) (5040 - 4900) ÷ (5184 - 4900) = 140 ÷ 284 ≈ 0.493

Step 3: 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 70 + 0.493 = 70.493, so the square root of 5040 is approximately 70.493.

• 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: The expression of a number as the product of its prime factors.

• Approximation method: A method used to find an estimated value of a quantity, often used when an exact answer is not possible or practical.