123 LearnersLast updated on May 26th, 2025
Square Root of 4689
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 4689.
What is the Square Root of 4689?
The square root is the inverse of the square of the number. 4689 is not a perfect square. The square root of 4689 can be expressed in both radical and exponential forms. In the radical form, it is expressed as √4689, whereas (4689)^(1/2) in the exponential form. √4689 ≈ 68.4653, 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 4689
The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where long division and approximation methods are used. Let us now learn the following methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 4689 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 4689 is broken down into its prime factors:
Step 1: Finding the prime factors of 4689
Breaking it down, we get 3 x 3 x 521: 3^2 x 521
Step 2: Now we found the prime factors of 4689. Since 4689 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √4689 using prime factorization is not straightforward.
Square Root of 4689 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 4689, we need to group it as 89 and 46.
Step 2: Now we need to find n whose square is ≤ 46. We can say n as ‘6’ because 6 x 6 = 36, which is less than 46. Subtract 36 from 46, and the remainder is 10.
Step 3: Bring down 89, which is the new dividend. Add the old divisor with the same number 6 + 6, we get 12, which will be our new divisor.
Step 4: The new divisor will have a placeholder for the next digit. Now we get 12_ as the new divisor, and we need to find the value of n.
Step 5: Find a digit for n where 12n x n ≤ 1089. Let us consider n as 8, now 128 x 8 = 1024.
Step 6: Subtract 1024 from 1089, the difference is 65, and the quotient is 68.
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 6500.
Step 8: Find the new divisor, which becomes 136 because 136 x 5 = 6800 is the closest without exceeding 6500.
Step 9: Subtracting 6800 from 6500 gives the remainder.
Step 10: The quotient becomes 68.46.
Step 11: Continue these steps until we get two numbers after the decimal point or when repeating steps yields a repeating pattern.
So, the square root of √4689 is approximately 68.4653.
Square Root of 4689 by Approximation Method
The approximation method is another method for finding square roots. It is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 4689 using the approximation method:
Step 1: Identify the closest perfect squares to √4689. The smallest perfect square less than 4689 is 4624, and the largest perfect square more than 4689 is 4761. √4689 falls somewhere between 68 and 69.
Step 2: Apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (4689 - 4624) / (4761 - 4624) = 65 / 137 ≈ 0.4745. Adding this decimal to the lower guess, we get 68 + 0.4745 = 68.4745. Therefore, √4689 ≈ 68.4745.
Common Mistakes and How to Avoid Them in the Square Root of 4689
Students often make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in the long division method. Let us look at a few common mistakes in detail.
Mistake 1
Forgetting about the negative square root
It is important to make students aware that a number has both positive and negative square roots. However, we usually take only the positive square root for practical applications.
For example, √50 = 7.07, but 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 students make. To resolve this, teach students to simplify numbers inside the square root first.
For example: √(12+24) = √36 is correct, not √12 + √24.
Mistake 3
Not finding the correct value of a non-perfect square
Teach students to simplify the numbers inside the square root to avoid mistakes.
For example, √50 ≠ 7; it is approximately 7.071067.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between cube root and square root 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. Students often skip a step accidentally, leading to wrong answers. This may occur during subtraction or elsewhere.
Square root of 4689 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √4689?
The area of the square is approximately 219,870.3 square units.
Explanation
The area of the square = side^2.
The side length is given as √4689.
Area of the square = side^2 = √4689 x √4689 = 4689.
Therefore, the area of the square box is approximately 219,870.3 square units.
Problem 2
A square-shaped building measuring 4689 square feet is built; if each of the sides is √4689, what will be the square feet of half of the building?
2344.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 4689 by 2 = 2344.5
So half of the building measures 2344.5 square feet.
Problem 3
Calculate √4689 x 5.
Approximately 342.3265
Explanation
The first step is to find the square root of 4689, which is approximately 68.4653.
The second step is to multiply 68.4653 by 5.
So, 68.4653 x 5 ≈ 342.3265
Problem 4
What will be the square root of (4689 + 25)?
The square root is approximately 70.
Explanation
To find the square root, first find the sum of (4689 + 25). 4689 + 25 = 4714, and then √4714 ≈ 68.65.
Therefore, the square root of (4689 + 25) is approximately 70.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √4689 units and the width ‘w’ is 50 units.
The perimeter of the rectangle is approximately 237.9306 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√4689 + 50) ≈ 2 × (68.4653 + 50) ≈ 2 × 118.4653 ≈ 237.9306 units.
FAQ on Square Root of 4689
1.What is √4689 in its simplest form?
2.Mention the factors of 4689.
3.Calculate the square of 4689.
4.Is 4689 a prime number?
5.4689 is divisible by?
6.How does learning Algebra help students in Indonesia make better decisions in daily life?
7.How can cultural or local activities in Indonesia support learning Algebra topics such as Square Root of 4689?
8.How do technology and digital tools in Indonesia support learning Algebra and Square Root of 4689?
9.Does learning Algebra support future career opportunities for students in Indonesia?
Important Glossaries for the Square Root of 4689
- 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 is used in practical applications, known as the principal square root.
- Prime factorization: The expression of a number as a product of its prime factors, such as 4689 = 3 x 3 x 521.
- Decimal approximation: The process of estimating an irrational number using decimal points for practical usage, such as √4689 ≈ 68.4653.
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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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