122 LearnersLast updated on May 26th, 2025
Square Root of 657
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, finance, etc. Here, we will discuss the square root of 657.
What is the Square Root of 657?
The square root is the inverse operation of squaring a number. Since 657 is not a perfect square, its square root is an irrational number. The square root of 657 is expressed in both radical and exponential forms. In radical form, it is expressed as √657, whereas in exponential form it is (657)^(1/2). The approximate value of √657 is 25.629, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
Finding the Square Root of 657
The prime factorization method is used for perfect square numbers, but not for non-perfect square numbers like 657. Instead, the long-division method or approximation method is used. Let us now explore these methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 657 by Prime Factorization Method
The prime factorization of a number involves breaking it down into the product of its prime factors. However, since 657 is not a perfect square, it cannot be simplified through prime factorization alone. Let's see its breakdown:
Step 1: Finding the prime factors of 657 Breaking it down, we get 3 x 3 x 73: 3^2 x 73
Step 2: Since 657 is not a perfect square, its prime factors cannot be perfectly paired.
Thus, calculating √657 using only prime factorization is not possible.
Square Root of 657 by Long Division Method
The long division method is particularly useful for non-perfect square numbers. Here’s how to find the square root of 657 using the long division method, step by step:
Step 1: Group the digits from right to left. For 657, we group it as 57 and 6.
Step 2: Find n such that n^2 is the largest perfect square less than or equal to 6. Here, n is 2 because 2^2 = 4.
Step 3: Subtract 4 from 6 to get a remainder of 2, and bring down 57, making the new dividend 257.
Step 4: Double the divisor (which is 2), and use it as the first part of the new divisor, getting 4n. Find n such that 4n × n is less than or equal to 257. Here, n is 5, since 45 × 5 = 225.
Step 5: Subtract 225 from 257 to get a remainder of 32. Extend the division by adding decimals and zeros as necessary.
Step 6: Continue repeating these steps until the desired precision is achieved.
Thus, the square root of 657 is approximately 25.629.
Square Root of 657 by Approximation Method
The approximation method is another way to estimate square roots. Here's how to approximate the square root of 657:
Step 1: Identify the closest perfect squares around 657. The closest perfect squares are 625 (25^2) and 676 (26^2).
Step 2: Since 657 is between these two perfect squares, √657 falls between 25 and 26.
Step 3: Use the formula: (Given number - smallest perfect square) / (Larger perfect square - smallest perfect square) Applying the formula: (657 - 625) / (676 - 625) = 32 / 51 ≈ 0.627 Add this decimal to 25, giving an approximation of 25.627 for √657.
Common Mistakes and How to Avoid Them in the Square Root of 657
Students often make mistakes when finding square roots, such as neglecting the negative square root or skipping steps in the long division method. Here are some common mistakes and how to avoid them:
Mistake 1
Forgetting about the negative square root
It's important to remember that every positive number has both a positive and a negative square root. While we usually focus on the positive root, the negative root should not be ignored.
For example: if √50 = 7.07, then -√50 = -7.07.
Mistake 2
Not adding the square root symbol in proper places
Misusing the square root symbol can lead to errors. It's crucial to simplify the expression under the square root before proceeding.
For example, √(12+24) ≠ √12 + √24. The correct form is √(12+24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
Simplifying the numbers inside the square root is essential to avoid errors in approximation.
For example, stating √50 = 7 is incorrect; the correct approximation is √50 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
Students should be clear on the distinction between square root (√) and cube root (∛).
For example, √50 and ∛50 are different operations.
Mistake 5
Making mistakes in long division
The long division method involves multiple steps and can lead to errors if steps are skipped. It's essential to carefully follow each step to ensure an accurate result.
Square root of 657 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √657?
The area of the square is approximately 657 square units.
Explanation
The area of a square = side^2.
The side length is given as √657.
Area of the square = side^2 = √657 × √657 = 657.
Therefore, the area of the square box is approximately 657 square units.
Problem 2
A square-shaped building measuring 657 square feet is built; if each of the sides is √657, what will be the square feet of half of the building?
328.5 square feet
Explanation
To find the area of half of the building, divide the given area by 2.
Dividing 657 by 2 gives us 328.5.
So half of the building measures 328.5 square feet.
Problem 3
Calculate √657 × 5.
Approximately 128.145
Explanation
First, find the square root of 657, which is approximately 25.629.
Multiply this by 5: 25.629 × 5 ≈ 128.145.
Problem 4
What will be the square root of (657 + 19)?
The square root is approximately 26.
Explanation
To find the square root, first calculate the sum of (657 + 19).
657 + 19 = 676.
The square root of 676 is 26, so the square root of (657 + 19) is ±26.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √657 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 127.258 units.
Explanation
Perimeter of a rectangle = 2 × (length + width).
Perimeter = 2 × (√657 + 38)
= 2 × (25.629 + 38)
≈ 2 × 63.629
= 127.258 units.
FAQ on Square Root of 657
1.What is √657 in its simplest form?
2.Mention the factors of 657.
3.Calculate the square of 657.
4.Is 657 a prime number?
5.657 is divisible by?
6.How does learning Algebra help students in Vietnam make better decisions in daily life?
7.How can cultural or local activities in Vietnam support learning Algebra topics such as Square Root of 657?
8.How do technology and digital tools in Vietnam support learning Algebra and Square Root of 657?
9.Does learning Algebra support future career opportunities for students in Vietnam?
Important Glossaries for the Square Root of 657
- Square root: A square root is the inverse of squaring a number. For example, 4^2 = 16, and the inverse is √16 = 4.
- Irrational number: An irrational number cannot be written as a simple fraction, where the denominator is not zero, and both the numerator and denominator are integers.
- Principal square root: The principal square root is the non-negative square root of a number, used in most real-world applications.
- Prime factorization: The process of expressing a number as the product of its prime factors. For example, 657 = 3 × 3 × 73.
- Long division method: A technique used to find the square root of a number by dividing the number into parts and estimating the square root iteratively.
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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.
Fun Fact
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