118 LearnersLast updated on May 26th, 2025
Square Root of 5040
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in the field of vehicle design, finance, etc. Here, we will discuss the square root of 5040.
What is the 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.
Finding the Square Root of 5040
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
Square Root of 5040 by Prime Factorization 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.
Square Root of 5040 by Long Division Method
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.
Square Root of 5040 by Approximation Method
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.
Common Mistakes and How to Avoid Them in the Square Root of 5040
Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping long division methods. Now let us look at a few of these mistakes that students tend to make in detail.
Mistake 1
Forgetting about the negative square root
It is important to make students aware that a number does have both positive and negative square roots. However, we will be taking only the positive square root, as it is the required one.
For example, if √50 ≈ 7.071, there is also -7.071, which should not be forgotten.
Mistake 2
Not adding the square root symbol in proper places
Not using the square root properly is another mistake that children make. To resolve this, we need to teach children that simplifying the numbers inside the square root is important and then move on to the second step.
For example, √(12+24) = √12 + √24 is the wrong method of using roots.
The right way of using roots is √(12+24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
The first step that needs to be taught to children is to simplify the numbers inside the square root. This is important so that students do not make mistakes while identifying the approximate value.
For example, saying √50 = 7 is wrong; the correct answer is √50 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between the cube root and square root constantly, as this will help them avoid mistakes.
For example, √50 and ∛50 are different.
Mistake 5
Making mistakes in long division
Finding a square root using the long division method involves many steps. Due to this, many students might skip a step accidentally, resulting in a wrong answer at the end of the calculation. This may happen during subtraction or elsewhere.
Square Root of 5040 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √5040?
The area of the square is approximately 5040 square units.
Explanation
The area of the square = side^2.
The side length is given as √5040.
Area of the square = (√5040)² ≈ 5040 square units.
Therefore, the area of the square box is approximately 5040 square units.
Problem 2
A square-shaped building measuring 5040 square feet is built; if each of the sides is √5040, what will be the square feet of half of the building?
2520 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 5040 by 2, we get 2520.
So half of the building measures 2520 square feet.
Problem 3
Calculate √5040 × 5.
Approximately 354.67
Explanation
The first step is to find the square root of 5040, which is approximately 70.933.
The second step is to multiply 70.933 by 5.
So 70.933 × 5 ≈ 354.67.
Problem 4
What will be the square root of (5040 + 60)?
The square root is approximately 71.063.
Explanation
To find the square root, sum (5040 + 60). 5040 + 60 = 5100, then √5100 ≈ 71.063.
Therefore, the square root of (5040 + 60) is approximately ±71.063.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √5040 units and the width ‘w’ is 40 units.
The perimeter of the rectangle is approximately 221.866 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√5040 + 40) ≈ 2 × (70.933 + 40) ≈ 2 × 110.933 ≈ 221.866 units.
FAQ on Square Root of 5040
1.What is √5040 in its simplest form?
2.Mention the factors of 5040.
3.Calculate the square of 5040.
4.Is 5040 a prime number?
5.5040 is divisible by?
6.How does learning Algebra help students in India make better decisions in daily life?
7.How can cultural or local activities in India support learning Algebra topics such as Square Root of 5040?
8.How do technology and digital tools in India support learning Algebra and Square Root of 5040?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of 5040
- 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.
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Jaskaran Singh Saluja
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Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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