117 LearnersLast updated on May 26th, 2025
Square Root of 698
If a number is multiplied by itself, the result is a square. The inverse operation is finding the square root. The square root is used in fields like vehicle design and finance. Here, we will discuss the square root of 698.
What is the Square Root of 698?
The square root is the inverse of squaring a number. Since 698 is not a perfect square, its square root is expressed in radical and exponential forms. In radical form, it is represented as √698, and in exponential form as (698)^(1/2). The approximate value of √698 is 26.41969, which is an irrational number because it cannot be expressed as a fraction of two integers.
Finding the Square Root of 698
For perfect squares, the prime factorization method is used. However, for non-perfect squares, methods like long division and approximation are employed. Let's explore these methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 698 by Prime Factorization Method
Prime factorization involves expressing a number as a product of its prime factors. Here's how 698 is broken down:
Step 1: Finding the prime factors of 698:
Breaking it down, we get 2 x 349: 2^1 x 349^1
Step 2: Since 698 is not a perfect square, the digits cannot be grouped into pairs. Thus, calculating the square root of 698 using prime factorization isn't possible.
Square Root of 698 by Long Division Method
The long division method is used for non-perfect squares. Here's how to find the square root using this method, step by step:
Step 1: Group the numbers from right to left. For 698, group it as 98 and 6.
Step 2: Find 'n' such that n^2 ≤ 6. n = 2 because 2 x 2 is less than or equal to 6. The quotient is 2, and the remainder is 2.
Step 3: Bring down 98 to make a new dividend of 298. Add 2 (old divisor) to itself to get 4, the new divisor.
Step 4: Find n such that 4n x n ≤ 298. Try n = 6, so 46 x 6 = 276.
Step 5: Subtract 276 from 298, leaving a remainder of 22. The quotient is 26.
Step 6: Since the remainder is less than the divisor, add a decimal point and two zeros to make the new dividend 2200.
Step 7: The new divisor is 529 because 529 x 4 = 2116, which is less than 2200.
Step 8: Subtract 2116 from 2200, leaving 84.
Step 9: The quotient is now 26.4. Continue these steps to get a more precise square root value.
The square root of √698 is approximately 26.42.
Square Root of 698 by Approximation Method
The approximation method finds square roots easily. Here’s how to approximate the square root of 698:
Step 1: Identify the closest perfect squares to 698. 676 and 729 are nearby perfect squares. √698 falls between 26 and 27.
Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). For 698, use (698 - 676) / (729 - 676) = 22 / 53 ≈ 0.415 Adding this to the smaller square root, we get 26 + 0.415 = 26.415.
So, the square root of 698 is approximately 26.42.
Common Mistakes and How to Avoid Them in the Square Root of 698
Students often make mistakes when finding square roots, such as forgetting about the negative square root or skipping steps in the long division method. Let's review common mistakes in detail.
Mistake 1
Forgetting About the Negative Square Root
Students should remember that a number has both positive and negative square roots. However, typically only the positive square root is considered.
For example: √50 = 7.07, but there is also -7.07.
Mistake 2
Not Using the Square Root Symbol Correctly
Incorrect use of the square root symbol is a common mistake. Students should simplify numbers inside the square root before proceeding.
Example: √(12+24) = √36, not √12 + √24.
Mistake 3
Miscalculating Non-Perfect Squares
Simplifying numbers inside the square root is crucial to avoid errors.
Example: √50 is not 7; the correct value is √50 ≈ 7.071067.
Mistake 4
Confusing Square Root with Cube Root
Teach students the difference between cube roots and square roots to avoid confusion.
For example: √50 and ∛50 are different.
Mistake 5
Errors in Long Division Steps
The long division method requires careful attention to detail. Skipping a step can result in incorrect answers. Ensure students understand each subtraction and calculation step.
Square Root of 698 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √698?
The area of the square is approximately 486.18 square units.
Explanation
The area of the square = side².
The side length is √698.
Area = (√698)² ≈ 26.42 × 26.42 = 698.
Therefore, the area of the square box is approximately 698 square units.
Problem 2
A square-shaped building measuring 698 square feet is built; if each of the sides is √698, what will be the square feet of half of the building?
349 square feet.
Explanation
Since the building is square-shaped, divide the given area by 2.
Dividing 698 by 2 gives 349.
So half of the building measures 349 square feet.
Problem 3
Calculate √698 × 5.
Approximately 132.1.
Explanation
First, find the square root of 698, which is approximately 26.42.
Then multiply 26.42 by 5: 26.42 × 5 ≈ 132.1
Problem 4
What will be the square root of (698 + 2)?
The square root is approximately 26.46.
Explanation
To find the square root, sum (698 + 2) to get 700.
√700 ≈ 26.46.
Therefore, the square root of (698 + 2) is approximately ±26.46.
Problem 5
Find the perimeter of a rectangle if its length 'l' is √698 units and the width 'w' is 38 units.
The perimeter of the rectangle is approximately 129.84 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√698 + 38) ≈ 2 × (26.42 + 38) ≈ 2 × 64.42 ≈ 128.84 units.
FAQ on Square Root of 698
1.What is √698 in its simplest form?
2.Mention the factors of 698.
3.Calculate the square of 698.
4.Is 698 a prime number?
5.698 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 698?
8.How do technology and digital tools in Vietnam support learning Algebra and Square Root of 698?
9.Does learning Algebra support future career opportunities for students in Vietnam?
Important Glossaries for the Square Root of 698
- Square root: The square root is the inverse of squaring a number. For example, 4² = 16, and the inverse is √16 = 4.
- Irrational number: An irrational number cannot be expressed as a simple fraction; for example, the square root of 698 is irrational.
- Decimal: A decimal is a number that includes a whole number and a fractional part, such as 26.42.
- Exponent: An exponent indicates how many times a number is multiplied by itself, such as ² in (698)^(1/2).
- Perimeter: The perimeter is the total distance around a two-dimensional shape, calculated by summing all side lengths.
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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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