122 LearnersLast updated on May 26th, 2025
Square Root of 578
If a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The square root is used in various fields such as engineering, physics, and finance. Here, we will discuss the square root of 578.
What is the Square Root of 578?
The square root is the inverse of the square of a number. 578 is not a perfect square. The square root of 578 is expressed in both radical and exponential forms. In the radical form, it is expressed as √578, whereas (578)^(1/2) is the exponential form. √578 ≈ 24.04163, 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 578
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers like 578, the long division method and approximation method are used. Let us now learn these methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 578 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 578 is broken down into its prime factors:
Step 1: Find the prime factors of 578. Breaking it down, we get 2 x 17 x 17: 2^1 x 17^2
Step 2: Now we found out the prime factors of 578. The second step is to make pairs of those prime factors. Since 578 is not a perfect square, the digits of the number can’t be grouped into pairs. Therefore, calculating the square root of 578 using prime factorization alone does not provide an exact result.
Square Root of 578 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 578, we need to group it as 78 and 5.
Step 2: Now we need to find n whose square is less than or equal to 5. We can say n is '2' because 2 x 2 = 4 is less than 5. Now the quotient is 2. After subtracting 4 from 5, the remainder is 1.
Step 3: Bring down 78, which is the new dividend. Add the old divisor with the same number 2 + 2, we get 4, which will be our new divisor.
Step 4: Find 4n such that 4n × n ≤ 178. Let us consider n as 4, now 44 x 4 = 176.
Step 5: Subtract 176 from 178, the difference is 2, and the quotient is 24.
Step 6: 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 200.
Step 7: Find the new divisor which is 48 because 480 x 0 = 0
Step 8: Subtract the product from 200 to continue the long division.
Step 9: Continue doing these steps until we get two numbers after the decimal point. If there is no decimal value, continue till the remainder is zero.
So the square root of √578 ≈ 24.04163.
Square Root of 578 by Approximation Method
The approximation method is another approach 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 578 using the approximation method.
Step 1: Now we have to find the closest perfect squares around √578. The smallest perfect square less than 578 is 576 and the largest perfect square greater than 578 is 625. √578 falls somewhere between √576 (24) and √625 (25).
Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (578 - 576) / (625 - 576) ≈ 0.04163. Using this formula, we identified the decimal point of our square root. The next step is adding the whole number we predicted initially to the decimal number: 24 + 0.04163 ≈ 24.04163, so the square root of 578 is approximately 24.04163.
Common Mistakes and How to Avoid Them in the Square Root of 578
Students do make mistakes while finding the square root, like forgetting about the negative square root, skipping steps in the long division method, etc. Now let us look at a few of these mistakes 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 = ±7.071, but we usually consider only the positive value.
Mistake 2
Not adding the square root symbol in proper places
Improper use of the square root symbol is another common mistake. To resolve this, we need to teach students that simplifying the numbers inside the square root is important before moving on to the next steps.
For example: √(12+24) ≠ √12 + √24. The correct approach is √(36) = 6.
Mistake 3
Not finding the correct value of a non-perfect square
The first step that needs to be taught is to simplify the numbers inside the square root. This is important so students do not make mistakes while identifying the approximate value.
For example: √50 ≠ 7; 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 roots and square roots to avoid confusion.
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, students might skip a step accidentally, resulting in a wrong answer. This may occur in subtraction or elsewhere in the process.
Square root of 578 Examples
Problem 1
Can you help Alex find the area of a square box if its side length is given as √578?
The area of the square is 578 square units.
Explanation
The area of the square = side^2.
The side length is given as √578.
Area of the square = (√578)^2 = 578 square units.
Therefore, the area of the square box is 578 square units.
Problem 2
A square-shaped garden measuring 578 square feet is built; if each of the sides is √578, what will be the square feet of half of the garden?
289 square feet
Explanation
We can just divide the given area by 2 as the garden is square-shaped.
Dividing 578 by 2 gives us 289. So half of the garden measures 289 square feet.
Problem 3
Calculate √578 x 3.
72.12489
Explanation
The first step is to find the square root of 578, which is approximately 24.04163.
The second step is to multiply 24.04163 by 3.
So 24.04163 x 3 ≈ 72.12489.
Problem 4
What will be the square root of (144 + 434)?
The square root is approximately 24.04163.
Explanation
To find the square root, we need to find the sum of (144 + 434). 144 + 434 = 578, and then √578 ≈ 24.04163.
Therefore, the square root of (144 + 434) is approximately 24.04163.
Problem 5
Find the perimeter of a rectangle if its length ‘l’ is √578 units and the width ‘w’ is 20 units.
The perimeter of the rectangle is approximately 88.08326 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√578 + 20) = 2 × (24.04163 + 20) ≈ 2 × 44.04163 = 88.08326 units.
FAQ on Square Root of 578
1.What is √578 in its simplest form?
2.Mention the factors of 578.
3.Calculate the square of 578.
4.Is 578 a prime number?
5.578 is divisible by?
6.How does learning Algebra help students in Bahrain make better decisions in daily life?
7.How can cultural or local activities in Bahrain support learning Algebra topics such as Square Root of 578?
8.How do technology and digital tools in Bahrain support learning Algebra and Square Root of 578?
9.Does learning Algebra support future career opportunities for students in Bahrain?
Important Glossaries for the Square Root of 578
- Square root: A square root is the inverse of a square. For 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, but the positive square root is more commonly used due to its applications in real-world problems. It is known as the principal square root.
- Prime factorization: The process of expressing a number as a product of its prime factors. For example, the prime factorization of 578 is 2 x 17 x 17.
- Long division method: A step-by-step method used to calculate the square root of a non-perfect square by dividing the number into groups of digits.
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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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