126 LearnersLast updated on May 26th, 2025
Square Root of 67
If a number is multiplied by itself, the result is a square. The inverse of squaring a number is finding its square root. Square roots are used in various fields such as vehicle design, finance, etc. Here, we will discuss the square root of 67.
What is the Square Root of 67?
The square root is the inverse operation of squaring a number. 67 is not a perfect square. The square root of 67 is expressed in both radical and exponential form.
In radical form, it is expressed as √67, whereas in exponential form it is expressed as (67)1/2. √67 ≈ 8.18535, which is an irrational number because it cannot be expressed as a fraction of two integers.
Finding the Square Root of 67
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers like 67, we use methods such as the long-division method and approximation method. Let's learn these methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 67 by Prime Factorization Method
Prime factorization involves expressing a number as a product of its prime factors. However, for non-perfect squares like 67, prime factorization alone does not help in finding the square root.
Step 1: Find the prime factors of 67. 67 is a prime number, so it can only be expressed as 67 x 1. Since 67 is not a perfect square, we cannot form pairs of its prime factors.
Therefore, calculating √67 using prime factorization is not feasible.
Square Root of 67 by Long Division Method
The long division method is suitable for non-perfect square numbers. This method involves finding the closest perfect square and proceeding step by step as follows:
Step 1: Start by grouping the numbers from right to left. For 67, consider it as 67.
Step 2: Find a number n whose square is less than or equal to 67. Here, n is 8, since 82 = 64, which is less than 67.
Step 3: Subtract 64 from 67, leaving a remainder of 3.
Step 4: Bring down pairs of zeros to the right of the remainder to get 300.
Step 5: Double the divisor (8), now 16, and guess the next digit of the quotient.
Step 6: Find the largest digit x such that 16x * x is less than or equal to 300.
Step 7: Continue this process to find more decimal places. The square root of √67 is approximately 8.18535.
Square Root of 67 by Approximation Method
The approximation method provides an easy way to estimate the square root of a number.
Step 1: Identify the two closest perfect squares. For 67, they are 64 (82) and 81 (92). √67 is between 8 and 9.
Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square).
Step 3: (67 - 64) / (81 - 64) = 3 / 17 ≈ 0.1765.
Step 4: Add this decimal to the smaller integer, 8, yielding 8 + 0.1765 ≈ 8.18.
Thus, the square root of 67 is approximately 8.18.
Common Mistakes and How to Avoid Them in the Square Root of 67
Students often make errors when finding square roots, such as forgetting the negative square root, skipping steps in the long division method, etc. Here are some common mistakes and how to avoid them:
Mistake 1
Forgetting about the negative square root
It's important to remember that every positive number has two square roots: one positive and one negative. However, typically only the principal (positive) square root is used in most applications. For example, √67 ≈ ±8.18535.
Square Root of 67 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √67?
The area of the square is approximately 67 square units.
Explanation
The area of the square = side2.
The side length is given as √67.
Area of the square = (√67)2 = 67.
Therefore, the area of the square box is approximately 67 square units.
Problem 2
A square-shaped building measuring 67 square feet is built; if each of the sides is √67, what will be the square feet of half of the building?
Approximately 33.5 square feet.
Explanation
To find half the area of the building,
divide the total area by 2. 67 / 2 = 33.5.
So, half of the building measures approximately 33.5 square feet.
Problem 3
Calculate √67 x 3.
Approximately 24.55605.
Explanation
First step is to find the square root of 67, which is approximately 8.18535. Then, multiply 8.18535 by 3. 8.18535 x 3 ≈ 24.55605.
Problem 4
What will be the square root of (62 + 5)?
The square root is approximately 8.18535.
Explanation
First, find the sum of (62 + 5). 62 + 5 = 67.
Then, √67 ≈ 8.18535.
Therefore, the square root of (62 + 5) is approximately 8.18535.
Problem 5
Find the perimeter of a rectangle if its length ‘l’ is √67 units and the width ‘w’ is 20 units.
The perimeter of the rectangle is approximately 56.37 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√67 + 20) ≈ 2 × (8.18535 + 20).
Perimeter ≈ 2 × 28.18535 ≈ 56.37 units.
FAQ on Square Root of 67
1.What is √67 in its simplest form?
2.What are the factors of 67?
3.Calculate the square of 67.
4.Is 67 a prime number?
5.67 is divisible by?
6.How does learning Algebra help students in United States make better decisions in daily life?
7.How can cultural or local activities in United States support learning Algebra topics such as Square Root of 67?
8.How do technology and digital tools in United States support learning Algebra and Square Root of 67?
9.Does learning Algebra support future career opportunities for students in United States?
Important Glossaries for the Square Root of 67
- Square Root: A square root is a value that, when multiplied by itself, gives the original number. Example: 42 = 16, and √16 = 4.
- Irrational Number: An irrational number cannot be expressed as a simple fraction; its decimal form is non-repeating and non-terminating.
- Principal Square Root: The principal square root is the positive square root of a number.
- Prime Number: A prime number has only two distinct factors: 1 and the number itself.
- Approximation: An approximation is a value or number that is close to the exact value, often used for simplifying calculations.
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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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