125 LearnersLast updated on May 26th, 2025
Square Root of 345
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 345.
What is the Square Root of 345?
The square root is the inverse of the square of the number. 345 is not a perfect square. The square root of 345 is expressed in both radical and exponential form. In radical form, it is expressed as √345, whereas (345)^(1/2) is the exponential form. √345 ≈ 18.5742, 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 345
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 method and approximation method are used. Let us now learn the following methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 345 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 345 is broken down into its prime factors:
Step 1: Finding the prime factors of 345 Breaking it down, we get 3 x 5 x 23: 3^1 x 5^1 x 23^1
Step 2: Now we found out the prime factors of 345. The second step is to make pairs of those prime factors. Since 345 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 345 using prime factorization is not straightforward.
Square Root of 345 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 345, we need to group it as 45 and 3.
Step 2: Now we need to find n whose square is closest to 3. We can say n as ‘1’ because 1 x 1 is lesser than or equal to 3. Now the quotient is 1, and after subtracting 1 from 3, the remainder is 2.
Step 3: Now let us bring down 45, which is the new dividend. Add the old divisor with the same number 1 + 1, we get 2, which will be our new divisor.
Step 4: The new divisor will be 2n, and we need to find the value of n.
Step 5: The next step is finding 2n x n ≤ 245. Let us consider n as 9; now, 29 x 9 = 261.
Step 6: Since 261 is greater than 245, we use n as 8. Now 28 x 8 = 224.
Step 7: Subtract 224 from 245; the difference is 21. The quotient is now 18.
Step 8: 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 2100.
Step 9: Now we need to find the new divisor. We get 369 when 369 x 5 = 1845.
Step 10: Subtracting 1845 from 2100, we get the result 255.
Step 11: Now the quotient is 18.5.
Step 12: Continue doing these steps until we get two numbers after the decimal point.
The square root of √345 ≈ 18.57.
Square Root of 345 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 345 using the approximation method.
Step 1: Now we have to find the closest perfect square of √345.
The smallest perfect square less than 345 is 324, and the largest perfect square greater than 345 is 361. √345 falls somewhere between 18 and 19.
Step 2: Now we need to apply the formula that is: (Given number - smaller perfect square) / (Greater perfect square - smaller perfect square) Going by the formula (345 - 324) ÷ (361 - 324) = 0.567
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: 18 + 0.567 = 18.567, so the square root of 345 is approximately 18.57.
Common Mistakes and How to Avoid Them in the Square Root of 345
Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in the long division method. 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 typically consider only the positive square root, as it is the one most often used.
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 symbol properly is another mistake that learners make. To resolve this, we need to teach learners that simplifying the numbers inside the square root is important before proceeding.
For Example: √(12 + 24) ≠ √12 + √24. The correct approach 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 learners is to simplify the numbers inside the square root. This is important so that they do not make mistakes while identifying the approximate value.
For Example: √50 ≈ 7.071, not 7.
Mistake 4
Confusing the square root symbol with the cube root
Learners need to be taught the difference between cube root and square root consistently, 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, learners might skip a step accidentally, resulting in a wrong answer at the end. This may happen during subtraction or elsewhere.
Square Root of 345 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √300?
The area of the square is 300 square units.
Explanation
The area of the square = side^2.
The side length is given as √300.
Area of the square = side^2 = √300 x √300 = 300.
Therefore, the area of the square box is 300 square units.
Problem 2
A square-shaped building measuring 345 square feet is built; if each of the sides is √345, what will be the square feet of half of the building?
172.5 square feet
Explanation
We can divide the given area by 2 as the building is square-shaped.
Dividing 345 by 2, we get 172.5.
So half of the building measures 172.5 square feet.
Problem 3
Calculate √345 x 5.
92.87
Explanation
The first step is to find the square root of 345, which is approximately 18.57.
The second step is to multiply 18.57 by 5.
So 18.57 x 5 ≈ 92.87.
Problem 4
What will be the square root of (300 + 45)?
The square root is 19.
Explanation
To find the square root, we need to find the sum of (300 + 45). 300 + 45 = 345. √345 ≈ 18.5742.
Therefore, the square root of (300 + 45) is approximately ±18.57.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √300 units and the width ‘w’ is 45 units.
We find the perimeter of the rectangle as 129.48 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√300 + 45) ≈ 2 × (17.32 + 45) = 2 × 62.32 = 124.64 units.
FAQ on Square Root of 345
1.What is √345 in its simplest form?
2.Mention the factors of 345.
3.Calculate the square of 345.
4.Is 345 a prime number?
5.345 is divisible by?
6.How does learning Algebra help students in Bahrain make better decisions in daily life?
7.How can cultural or local activities in Bahrain support learning Algebra topics such as Square Root of 345?
8.How do technology and digital tools in Bahrain support learning Algebra and Square Root of 345?
9.Does learning Algebra support future career opportunities for students in Bahrain?
Important Glossaries for the Square Root of 345
- 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 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 often the positive square root that is emphasized due to its usefulness in real-world applications. This is why it is also known as the principal square root.
- Prime factorization: The process of expressing a number as the product of its prime factors. For example, 345 = 3 x 5 x 23.
- Decimal: A number that includes a whole number and a fractional part, separated by a decimal point. For example: 7.86, 8.65, and 9.42 are decimal numbers.
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About BrightChamps in Bahrain
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.
