118 LearnersLast updated on May 26th, 2025
Square Root of 688
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 fields like vehicle design, finance, etc. Here, we will discuss the square root of 688.
What is the Square Root of 688?
The square root is the inverse of the square of the number. 688 is not a perfect square. The square root of 688 is expressed in both radical and exponential form. In the radical form, it is expressed as √688, whereas (688)^(1/2) in the exponential form. √688 ≈ 26.229, which is an irrational number because it cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.
Finding the Square Root of 688
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 methods and approximation methods are used. Let us now learn the following methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 688 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 688 is broken down into its prime factors.
Step 1: Finding the prime factors of 688
Breaking it down, we get 2 x 2 x 2 x 43: 2^3 x 43^1
Step 2: Now we found out the prime factors of 688. The second step is to make pairs of those prime factors. Since 688 is not a perfect square, the digits of the number can’t be grouped in a pair. Therefore, calculating 688 using prime factorization to find a whole number is impossible.
Square Root of 688 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 688, we need to group it as 88 and 6.
Step 2: Now we need to find n whose square is 6. We can say n as ‘2’ because 2 x 2 is lesser than or equal to 6. Now the quotient is 2, and after subtracting 4 from 6, the remainder is 2.
Step 3: Now let us bring down 88, which is the new dividend. Add the old divisor with the same number 2 + 2 to get 4, which will be our new divisor.
Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 4n as the new divisor, and we need to find the value of n.
Step 5: The next step is finding 4n x n ≤ 288. Let us consider n as 6; now 46 x 6 = 276.
Step 6: Subtracting 276 from 288, the difference is 12, and the quotient is 26.
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 1200.
Step 8: Now we need to find the new divisor, which is 524 because 524 x 2 = 1048.
Step 9: Subtracting 1048 from 1200, we get the result 152.
Step 10: Now the quotient is 26.2.
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.
So the square root of √688 is approximately 26.23.
Square Root of 688 by Approximation Method
The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 688 using the approximation method.
Step 1: Now we have to find the closest perfect square of √688. The smallest perfect square less than 688 is 676, and the largest perfect square more than 688 is 729. √688 falls somewhere between 26 and 27.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula (688 - 676) ÷ (729 - 676) = 12 ÷ 53 ≈ 0.23. Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 26 + 0.23 = 26.23, so the square root of 688 is approximately 26.23.
Common Mistakes and How to Avoid Them in the Square Root of 688
Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping long division methods. Now let us look at a few of those mistakes that students tend to make in detail.
Mistake 1
Forgetting about the negative square root
It is important to make students aware that a number does have both positive and negative square roots. However, we will be taking only the positive square root, as it is the required one.
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 that children make. To resolve this, we need to teach children that simplifying the numbers inside the square root is important and then move on to the second step.
For example, √(12+24) = √12 + √24 is the wrong method of using roots.
The right way of using roots is √(12+24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
The first step that needs to be taught to children is to simplify the numbers inside the square root. This is important so that students do not make mistakes while identifying the approximate value.
For example, √50 = 7 is wrong; the correct answer is √50 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between cube root and square root constantly, as this will help them avoid mistakes.
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. Due to this, many students might skip a step accidentally, which results in a wrong answer at the end of the sum. This may either happen in subtracting or elsewhere.
Square Root of 688 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √688?
The area of the square is approximately 688 square units.
Explanation
The area of the square = side^2.
The side length is given as √688.
Area of the square = side^2 = √688 x √688 = 688.
Therefore, the area of the square box is approximately 688 square units.
Problem 2
A square-shaped building measuring 688 square feet is built. If each of the sides is √688, what will be the square feet of half of the building?
344 square feet
Explanation
We can just divide the given area by 2, as the building is square-shaped.
Dividing 688 by 2, we get 344.
So half of the building measures 344 square feet.
Problem 3
Calculate √688 x 5.
Approximately 131.145
Explanation
The first step is to find the square root of 688, which is approximately 26.229.
The second step is to multiply 26.229 by 5.
So 26.229 x 5 ≈ 131.145.
Problem 4
What will be the square root of (650 + 38)?
The square root is approximately 26.229.
Explanation
To find the square root, we need to find the sum of (650 + 38). 650 + 38 = 688, and then √688 ≈ 26.229.
Therefore, the square root of (650 + 38) is approximately 26.229.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √688 units and the width ‘w’ is 20 units.
We find the perimeter of the rectangle as approximately 92.458 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√688 + 20) ≈ 2 × (26.229 + 20) ≈ 2 × 46.229 ≈ 92.458 units.
FAQ on Square Root of 688
1.What is √688 in its simplest form?
2.Mention the factors of 688.
3.Calculate the square of 688.
4.Is 688 a prime number?
5.688 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 688?
8.How do technology and digital tools in Vietnam support learning Algebra and Square Root of 688?
9.Does learning Algebra support future career opportunities for students in Vietnam?
Important Glossaries for the Square Root of 688
- 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 has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
- Prime factorization: This is the process of expressing a number as a product of its prime factors.
- Long division method: A method used to find the square root of a non-perfect square by dividing the number into groups and solving step by step.
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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.
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