Square Root of 698
2026-02-28 17:26 Diff
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Last updated on August 5, 2025

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.

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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.

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Problem 1

Can you help Max find the area of a square box if its side length is given as √698?

Okay, lets begin

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.

Well explained 👍

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?

Okay, lets begin

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.

Well explained 👍

Problem 3

Calculate √698 × 5.

Okay, lets begin

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

Well explained 👍

Problem 4

What will be the square root of (698 + 2)?

Okay, lets begin

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.

Well explained 👍

Problem 5

Find the perimeter of a rectangle if its length 'l' is √698 units and the width 'w' is 38 units.

Okay, lets begin

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.

Well explained 👍

FAQ on Square Root of 698

1.What is √698 in its simplest form?

The prime factorization of 698 is 2 × 349, so the simplest form of √698 is √(2 × 349).

2.Mention the factors of 698.

Factors of 698 are 1, 2, 349, and 698.

3.Calculate the square of 698.

To find the square of 698, multiply the number by itself: 698 × 698 = 487204.

4.Is 698 a prime number?

698 is not a prime number because it has more than two factors.

5.698 is divisible by?

698 is divisible by 1, 2, 349, and 698.

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

: He loves to play the quiz with kids through algebra to make kids love it.