112 LearnersLast updated on May 26th, 2025
Square Root of 5648
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 fields such as vehicle design, finance, etc. Here, we will discuss the square root of 5648.
What is the Square Root of 5648?
The square root is the inverse of the square of the number. 5648 is not a perfect square. The square root of 5648 is expressed in both radical and exponential form. In the radical form, it is expressed as √5648, whereas (5648)¹/² in the exponential form. √5648 ≈ 75.146, 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 5648
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 5648 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 5648 is broken down into its prime factors.
Step 1: Finding the prime factors of 5648 Breaking it down, we get 2 x 2 x 2 x 2 x 353: 2⁴ x 353
Step 2: Now we found out the prime factors of 5648. The second step is to make pairs of those prime factors. Since 5648 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 5648 using prime factorization is impossible.
Square Root of 5648 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 5648, we need to group it as 48 and 56.
Step 2: Now we need to find n whose square is closest to or less than 56. We can use n as ‘7’ because 7 x 7 is 49, which is less than 56. Now the quotient is 7, and after subtracting 49 from 56, the remainder is 7.
Step 3: Now, let us bring down 48, which is the new dividend. Add the old divisor with the same number (7 + 7) to get 14, which will be our new divisor.
Step 4: The new divisor will be the sum of the current divisor and the next digit of the quotient. Now we get 14n as the new divisor, and we need to find the value of n.
Step 5: The next step is to find 14n x n ≤ 748. Let us consider n as 5 since 145 x 5 = 725, which is closest to 748.
Step 6: Subtract 725 from 748; the difference is 23, and the quotient is 75.
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 2300.
Step 8: Now we need to find the new divisor. We find 150n x n ≤ 2300. Let us consider n as 1 since 1501 x 1 = 1501.
Step 9: Subtracting 1501 from 2300, we get 799.
Step 10: Continue doing these steps until we get two numbers after the decimal point. Suppose there are no decimal values; continue till the remainder is zero.
So the square root of √5648 ≈ 75.146.
Square Root of 5648 by Approximation Method
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 5648 using the approximation method.
Step 1: Now we have to find the closest perfect square of √5648.
The smallest perfect square less than 5648 is 5476 (74²), and the largest perfect square greater than 5648 is 5776 (76²). √5648 falls somewhere between 74 and 76.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Using the formula (5648 - 5476) ÷ (5776 - 5476) ≈ 0.146.
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 74 + 0.146 ≈ 74.146.
So the square root of 5648 is approximately 75.146.
Common Mistakes and How to Avoid Them in the Square Root of 5648
Students do make mistakes while finding the square root, like forgetting about the negative square root, skipping long division methods, etc. Now let us look at a few of those 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: √50 = 7.07, there is also -7.07 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. 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: √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 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 either happen in subtracting or elsewhere.
Square Root of 5648 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √5648?
The area of the square box is 5648 square units.
Explanation
The area of the square = side².
The side length is given as √5648.
Area of the square = side² = √5648 x √5648 = 5648.
Therefore, the area of the square box is 5648 square units.
Problem 2
A square-shaped field measuring 5648 square meters is built; if each of the sides is √5648, what will be the area of half of the field?
2824 square meters.
Explanation
We can just divide the given area by 2 as the field is square-shaped.
Dividing 5648 by 2 gives us 2824.
So half of the field measures 2824 square meters.
Problem 3
Calculate √5648 x 2.
150.292
Explanation
The first step is to find the square root of 5648, which is approximately 75.146.
The second step is to multiply 75.146 by 2.
So 75.146 x 2 = 150.292.
Problem 4
What will be the square root of (5476 + 172)?
The square root is approximately 75.
Explanation
To find the square root, we need to find the sum of (5476 + 172). 5476 + 172 = 5648, and then √5648 ≈ 75.146.
Therefore, the square root of (5476 + 172) is approximately ±75.146.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √5648 units and the width ‘w’ is 48 units.
We find the perimeter of the rectangle as approximately 246.292 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√5648 + 48) = 2 × (75.146 + 48) = 2 × 123.146 = 246.292 units.
FAQ on Square Root of 5648
1.What is √5648 in its simplest form?
2.Mention the factors of 5648.
3.Calculate the square of 5648.
4.Is 5648 a prime number?
5.5648 is divisible by?
6.How does learning Algebra help students in Australia make better decisions in daily life?
7.How can cultural or local activities in Australia support learning Algebra topics such as Square Root of 5648?
8.How do technology and digital tools in Australia support learning Algebra and Square Root of 5648?
9.Does learning Algebra support future career opportunities for students in Australia?
Important Glossaries for the Square Root of 5648
- 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 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.
- Approximation method: A mathematical approach used to find a close estimate of the square root for non-perfect squares.
- Long division method: A method used to find the square root of a number by dividing it into pairs of digits and using iterative division steps.
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Jaskaran Singh Saluja
About the Author
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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