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.

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

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.

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.

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.

• 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.