116 LearnersLast updated on May 26th, 2025
Square Root of 306
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 306.
What is the Square Root of 306?
The square root is the inverse of the square of the number. 306 is not a perfect square. The square root of 306 is expressed in both radical and exponential form. In the radical form, it is expressed as √306, whereas (306)^(1/2) in the exponential form. √306 ≈ 17.49285, 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 306
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 306 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 306 is broken down into its prime factors.
Step 1: Finding the prime factors of 306
Breaking it down, we get 2 x 3 x 3 x 17: 2^1 x 3^2 x 17^1
Step 2: Now we found out the prime factors of 306. The second step is to make pairs of those prime factors. Since 306 is not a perfect square, therefore the digits of the number can’t be grouped in pair. Therefore, calculating 306 using prime factorization is not straightforward.
Square Root of 306 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 306, we need to group it as 06 and 3.
Step 2: Now we need to find n whose square is less than or equal to 3. We can say n is '1' because 1 x 1 is lesser than or equal to 3. Now the quotient is 1. After subtracting 1 from 3, the remainder is 2.
Step 3: Now let us bring down 06 which is the new dividend. Add the old divisor with the same number 1 + 1 to get 2, which will be our new divisor.
Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, we need to find the value of n.
Step 5: The next step is finding 2n x n ≤ 206. Let us consider n as 8. Now 28 x 8 = 224, which is too large. Trying n as 7 works since 27 x 7 = 189.
Step 6: Subtract 189 from 206. The difference is 17, and the quotient is 17.
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 1700.
Step 8: Now we need to find the new divisor. If the new divisor is 354, then 354 x 4 = 1416, which fits into 1700.
Step 9: Subtracting 1416 from 1700, we get the result 284.
Step 10: Now the quotient is 17.4.
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 √306 is 17.49.
Square Root of 306 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 306 using the approximation method.
Step 1: Now we have to find the closest perfect squares of √306. The closest perfect square less than 306 is 289, and the closest perfect square greater than 306 is 324. √306 falls somewhere between 17 and 18.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Using the formula (306 - 289) ÷ (324 - 289) = 17 ÷ 35 ≈ 0.486. 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 17 + 0.49 = 17.49. 17 + 0.49 = 17.49, so the square root of 306 is approximately 17.49.
Common Mistakes and How to Avoid Them in the Square Root of 306
Students do make mistakes while finding the square root, like forgetting about the negative square root or skipping long division methods, etc. 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 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 a 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 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 306 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √306?
The area of the square is approximately 306 square units.
Explanation
The area of the square = side^2.
The side length is given as √306.
Area of the square = side^2 = √306 x √306 = 306.
Therefore, the area of the square box is approximately 306 square units.
Problem 2
A square-shaped building measuring 306 square feet is built; if each of the sides is √306, what will be the square feet of half of the building?
153 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 306 by 2, we get 153.
So half of the building measures 153 square feet.
Problem 3
Calculate √306 x 5.
Approximately 87.46
Explanation
The first step is to find the square root of 306 which is approximately 17.49, the second step is to multiply 17.49 by 5.
So 17.49 x 5 ≈ 87.46.
Problem 4
What will be the square root of (256 + 50)?
The square root is 18.
Explanation
To find the square root, we need to find the sum of (256 + 50). 256 + 50 = 306, and then √306 ≈ 17.49.
This is a slight error.
Let's take a proper example: 256 + 50 = 306, which is not a perfect square, but √306 ≈ 17.49.
Therefore, the square root of (256 + 50) is approximately 17.49.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √306 units and the width ‘w’ is 20 units.
The perimeter of the rectangle is approximately 74.98 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√306 + 20) = 2 × (17.49 + 20) = 2 × 37.49 ≈ 74.98 units.
FAQ on Square Root of 306
1.What is √306 in its simplest form?
2.Mention the factors of 306.
3.Calculate the square of 306.
4.Is 306 a prime number?
5.306 is divisible by?
6.How does learning Algebra help students in Philippines make better decisions in daily life?
7.How can cultural or local activities in Philippines support learning Algebra topics such as Square Root of 306?
8.How do technology and digital tools in Philippines support learning Algebra and Square Root of 306?
9.Does learning Algebra support future career opportunities for students in Philippines?
Important Glossaries for the Square Root of 306
- 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.
- Prime factorization: Expressing a number as a product of its prime factors. For example, 306 = 2 x 3 x 3 x 17.
- Approximation method: A technique to find an approximate value for the square root of a number by using nearby perfect squares.
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Jaskaran Singh Saluja
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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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