123 LearnersLast updated on May 26th, 2025
Square Root of 178
If a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The square root is used in fields such as vehicle design, finance, etc. Here, we will discuss the square root of 178.
What is the Square Root of 178?
The square root is the inverse of the square of the number. 178 is not a perfect square. The square root of 178 is expressed in both radical and exponential form. In radical form, it is expressed as √178, whereas (178)^(1/2) in exponential form. √178 ≈ 13.34166, 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 178
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers like 178, methods such as the long-division method and approximation method are used. Let us now learn these methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 178 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 178 is broken down into its prime factors:
Step 1: Finding the prime factors of 178 Breaking it down, we get 2 × 89: 2^1 × 89^1
Step 2: Now we have found the prime factors of 178. Since 178 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 178 using prime factorization alone is not feasible for finding an exact square root.
Square Root of 178 by Long Division Method
The long division method is particularly used for non-perfect square numbers. In this method, we find the square root using a step-by-step division process.
Step 1: To begin with, we need to group the numbers from right to left. In the case of 178, we group it as 78 and 1.
Step 2: Now, find a number n whose square is less than or equal to 1. We use n = 1 because 1 × 1 ≤ 1. The quotient is 1, and after subtracting, the remainder is 0.
Step 3: Bring down 78 to form the new dividend. Add the old divisor with the same number: 1 + 1 = 2, which becomes the new divisor.
Step 4: The new divisor is 2n. We need to find a value for n such that 2n × n ≤ 78. Trying n = 3, we have 23 × 3 = 69.
Step 5: Subtracting 69 from 78 gives a remainder of 9. The quotient is 13.
Step 6: Since the dividend is less than the divisor, we add a decimal point and continue the process by bringing down pairs of zeros.
Step 7: Continue this process until we have enough decimal places. So the square root of √178 ≈ 13.341.
Square Root of 178 by Approximation Method
The approximation method is another way to find square roots. It is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 178 using the approximation method.
Step 1: Find the closest perfect squares surrounding 178. The closest perfect squares are 169 (13^2) and 196 (14^2). So √178 falls between 13 and 14.
Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). (178 - 169)/(196 - 169) = 9/27 ≈ 0.333 Using the formula, we identify the decimal point of our square root. The next step is adding the value we found initially to the decimal number: 13 + 0.333 ≈ 13.333, so the square root of 178 is approximately 13.333.
Common Mistakes and How to Avoid Them in the Square Root of 178
Students often make mistakes while finding square roots, such as forgetting about the negative square root, skipping long division methods, etc. Let's look at a few common mistakes in detail.
Mistake 1
Forgetting about the negative square root
It is important to remember that a number has both positive and negative square roots. However, we usually consider only the positive square root in most practical applications.
For example: √50 ≈ 7.07, but there is also -7.07, which should not be ignored.
Mistake 2
Not adding the square root symbol in proper places
Misplacing the square root symbol is a common mistake. Teach students to simplify the numbers inside the square root before proceeding.
For example: √(12 + 24) = √36 is correct, whereas √12 + √24 is incorrect.
Mistake 3
Not finding the correct value of a non-perfect square
Teach students to simplify numbers inside the square root to avoid errors in approximation.
For example: √50 ≈ 7.071, not 7.
Mistake 4
Confusing the square root symbol with the cube root
Distinguish between square roots and cube roots.
For example, √50 and ∛50 are different operations.
Mistake 5
Making mistakes in long division
Finding a square root using the long division method involves many steps. Students might skip a step, leading to incorrect results. Ensure careful execution of each step, especially in subtraction and bringing down pairs of numbers.
Square Root of 178 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √178?
The area of the square is approximately 178 square units.
Explanation
The area of the square = side².
The side length is given as √178.
Area = (√178)² = 178.
Therefore, the area of the square box is approximately 178 square units.
Problem 2
A square-shaped building measuring 178 square feet is built; if each of the sides is √178, what will be the square feet of half of the building?
89 square feet
Explanation
Since the building is square-shaped, dividing the total area by 2 gives the area of half the building. 178 ÷ 2 = 89.
So half of the building measures 89 square feet.
Problem 3
Calculate √178 × 5.
Approximately 66.705
Explanation
First, find the square root of 178, which is approximately 13.341.
Then multiply 13.341 by 5. So, 13.341 × 5 ≈ 66.705.
Problem 4
What will be the square root of (178 + 22)?
The square root is 14.
Explanation
First, find the sum of (178 + 22). 178 + 22 = 200, and then find the square root of 200. √200 ≈ 14.14.
Therefore, the square root of (178 + 22) is approximately ±14.14.
Problem 5
Find the perimeter of a rectangle if its length ‘l’ is √178 units and the width ‘w’ is 40 units.
The perimeter of the rectangle is approximately 106.682 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√178 + 40) ≈ 2 × (13.341 + 40) ≈ 2 × 53.341 ≈ 106.682 units.
FAQ on Square Root of 178
1.What is √178 in its simplest form?
2.Mention the factors of 178.
3.Calculate the square of 178.
4.Is 178 a prime number?
5.178 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 178?
8.How do technology and digital tools in Bahrain support learning Algebra and Square Root of 178?
9.Does learning Algebra support future career opportunities for students in Bahrain?
Important Glossaries for the Square Root of 178
- Square root: A square root is the inverse operation to squaring a number. For example, 4^2 = 16, and the inverse is the square root: √16 = 4.
- Irrational number: A number that cannot be expressed as a simple fraction, such as √178, is an irrational number.
- Principal square root: The principal square root is the non-negative square root of a number. For example, the principal square root of 178 is √178 ≈ 13.341.
- Prime factorization: Breaking down a number into its prime factors, such as 178 = 2 × 89.
- Long division method: A method to find the square root of non-perfect squares through a division-like process that involves multiple steps.
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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.
