121 LearnersLast updated on May 26th, 2025
Square Root of 706
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 the field of vehicle design, finance, etc. Here, we will discuss the square root of 706.
What is the Square Root of 706?
The square root is the inverse of the square of the number. 706 is not a perfect square. The square root of 706 is expressed in both radical and exponential form. In the radical form, it is expressed as √706, whereas (706)^(1/2) in the exponential form. √706 ≈ 26.57066, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.
Finding the Square Root of 706
The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where the long-division method and approximation method are used. Let us now learn the following methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 706 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 706 is broken down into its prime factors.
Step 1: Finding the prime factors of 706
Breaking it down, we get 2 x 353: 2^1 x 353^1
Step 2: Now that we have found the prime factors of 706, the second step is to make pairs of those prime factors. Since 706 is not a perfect square, the digits of the number can't be grouped in pairs. Therefore, calculating 706 using prime factorization is not straightforward.
Square Root of 706 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 706, we need to group it as 06 and 7.
Step 2: Now we need to find n whose square is ≤ 7. We can say n is ‘2’ because 2 x 2 = 4 which is less than 7. Now the quotient is 2, after subtracting 4 from 7, the remainder is 3.
Step 3: Now let us bring down 06, which is the new dividend. Add the old divisor with the same number 2 + 2, we 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, we need to find the value of n.
Step 5: The next step is finding 4n × n ≤ 306. Let us consider n as 7, now 47 x 7 = 329, which is too much, so we try n as 6.
Step 6: 46 x 6 = 276. Subtract 276 from 306, the difference is 30, 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 zeros to the dividend. Now the new dividend is 3000.
Step 8: Now we need to find the new divisor that is 532 because 532 x 5 = 2660.
Step 9: Subtracting 2660 from 3000, we get the result 340.
Step 10: Now the quotient is 26.5.
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.
So the square root of √706 is approximately 26.57.
Square Root of 706 by Approximation Method
The approximation method is another method for finding the 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 706 using the approximation method.
Step 1: Now we have to find the closest perfect square of √706. The smallest perfect square less than 706 is 676 and the largest perfect square more than 706 is 729. √706 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 (706 - 676) ÷ (729 - 676) = 30 ÷ 53 ≈ 0.566 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 + 0.57 = 26.57, so the square root of 706 is approximately 26.57.
Common Mistakes and How to Avoid Them in the Square Root of 706
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, 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 ≈ 7.071067.
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 keep away from 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 706 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √706?
The area of the square is approximately 706 square units.
Explanation
The area of the square = side².
The side length is given as √706.
Area of the square = side² = √706 x √706 = 706
Therefore, the area of the square box is approximately 706 square units.
Problem 2
A square-shaped building measuring 706 square feet is built; if each of the sides is √706, what will be the square feet of half of the building?
353 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 706 by 2, we get 353.
So, half of the building measures 353 square feet.
Problem 3
Calculate √706 x 5.
Approximately 132.85
Explanation
The first step is to find the square root of 706, which is approximately 26.57. T
he second step is to multiply 26.57 by 5.
So, 26.57 x 5 ≈ 132.85
Problem 4
What will be the square root of (676 + 30)?
The square root is 26.
Explanation
To find the square root, we need to find the sum of (676 + 30). 676 + 30 = 706, and then √706 ≈ 26.57, which rounds to 26.
Therefore, the square root of (676 + 30) is approximately ±26.57.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √706 units and the width ‘w’ is 50 units.
We find the perimeter of the rectangle as approximately 153.14 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√706 + 50) = 2 × (26.57 + 50) = 2 × 76.57 ≈ 153.14 units.
FAQ on Square Root of 706
1.What is √706 in its simplest form?
2.Mention the factors of 706.
3.Calculate the square of 706.
4.Is 706 a prime number?
5.706 is divisible by?
6.How does learning Algebra help students in Singapore make better decisions in daily life?
7.How can cultural or local activities in Singapore support learning Algebra topics such as Square Root of 706?
8.How do technology and digital tools in Singapore support learning Algebra and Square Root of 706?
9.Does learning Algebra support future career opportunities for students in Singapore?
Important Glossaries for the Square Root of 706
- 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 a principal square root.
- Long Division Method: A method used to find the square root of non-perfect square numbers by dividing and finding the quotient and remainder step by step.
- Approximation Method: A method used to estimate the square root by finding the closest perfect squares and determining a decimal value.
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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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