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139 LearnersLast updated on December 15, 2025
Square Root of 64/9
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 various fields such as vehicle design and finance. Here, we will discuss the square root of 64/9.
What is the Square Root of 64/9?
The square root is the inverse of the square of the number. 64/9 is a perfect square.
The square root of 64/9 is expressed in both radical and exponential form.
In the radical form, it is expressed as √(64/9), whereas (64/9)^(1/2) in the exponential form.
√(64/9) = 8/3, which is a rational number because it can be expressed in the form of p/q, where p and q are integers and q ≠ 0.
Finding the Square Root of 64/9
The prime factorization method for perfect square numbers is straightforward.
For fractions, the square root is calculated by finding the square root of the numerator and the denominator separately.
- Prime factorization method
- Long division method
- Rationalization method
Square Root of 64/9 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 64/9 can be broken down:
Step 1: Find the prime factors of the numerator and the denominator. 64 = 2 × 2 × 2 × 2 × 2 × 2 = 26 9 = 3 × 3 = 3^2
Step 2: Take the square root of both the numerator and the denominator. √64 = 2(6/2) = 23 = 8 √9 = 3(2/2) = 3 Therefore, the square root of 64/9, √(64/9) = 8/3.
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Square Root of 64/9 by Long Division Method
The long division method is used for precise calculation of non-perfect square numbers, but for 64/9, this is straightforward due to its being a perfect square.
However, if needed for other numbers:
Step 1: Use the long division separately for 64 and 9 to find their square roots.
Step 2: Divide the square root of the numerator by the square root of the denominator. In this case: Square root of 64 is 8. Square root of 9 is 3. Thus, √(64/9) = 8/3.
Square Root of 64/9 by Rationalization Method
Rationalization is useful when dealing with square roots in the denominator.
Step 1: For √(64/9), it's already rational, but if needed for others, multiply by a form of 1 that will clear the root from the denominator.
Step 2: Simplify. For √(64/9), since it's already rational as 8/3, no further steps are needed.
Common Mistakes and How to Avoid Them in the Square Root of 64/9
Students may make mistakes while finding the square root, such as forgetting about the negative square root or misapplying methods.
Let us look at a few mistakes in detail.
Mistake 1
Forgetting about the negative square root
It's important to remember that every positive number has both a positive and negative square root.
However, we often focus on the principal (positive) square root in practical applications.
For example: √(64/9) = ±8/3.
Mistake 2
Incorrect simplification of fractions
When working with fractions under a square root, ensure each part is simplified correctly.
For instance, √(64/9) should be calculated as √64/√9 = 8/3, not by incorrectly simplifying terms.
Mistake 3
Misidentifying non-perfect squares as perfect squares
Sometimes, students may incorrectly assume a non-perfect square is perfect.
Verifying whether both the numerator and denominator are perfect squares can help avoid this mistake.
Mistake 4
Confusing square root with other operations
Students should clearly distinguish between square roots and other operations, such as cube roots or arithmetic operations like division.
Mistake 5
Errors in long division steps
While long division is not necessary for perfect squares, ensuring each step is followed correctly is crucial for other numbers to prevent errors.
Square Root of 64/9 Examples
Problem 1
Can you help Max find the side length of a square box if its area is given as 64/9 square units?
The side length of the square box is 8/3 units.
Explanation
The side length of a square is the square root of its area.
Area of the square = side2 = 64/9.
Thus, side = √(64/9) = 8/3.
Problem 2
A square-shaped plot measures 64/9 square feet; what is the perimeter if each of the sides is โ(64/9)?
32/3 feet.
Explanation
The perimeter of a square is 4 times the length of one side.
Side length = √(64/9) = 8/3.
So, perimeter = 4 × (8/3) = 32/3 feet.
Problem 3
Calculate the value of โ(64/9) ร 3.
8
Explanation
First, find the square root of 64/9, which is 8/3.
Then multiply by 3: (8/3) × 3 = 8.
Problem 4
What will be the square root of (64/9) + (16/9)?
The square root is 4/3.
Explanation
To find the square root, first add: (64/9) + (16/9) = (80/9).
The square root of 80/9 is √(80/9) = 4√5/3.
Since it's a complex result, we simplify it to find the closest perfect square: 64/9, which is (8/3).
Therefore, the closest perfect square approximation is √(16/9) = 4/3.
Problem 5
Find the area of a rectangle if its length โlโ is โ(64/9) units and the width โwโ is 3 units.
We find the area of the rectangle is 8 square units.
Explanation
Area of the rectangle = length × width Area = √(64/9) × 3 = (8/3) × 3 = 8 square units.
FAQ on Square Root of 64/9
1.What is โ(64/9) in its simplest form?
2.Mention the factors of 64/9.
3.Calculate the square of 64/9.
4.Is 64/9 a rational number?
5.64/9 is divisible by?
Important Glossaries for the Square Root of 64/9
- Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, √(64/9) = 8/3.
- Rational number: A number that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero.
- Perfect square: A number that is the square of an integer. For example, 64 is a perfect square because 8 × 8 = 64.
- Fraction: A numerical quantity that is not a whole number, representing a part of a whole, expressed as 'a/b' where 'b' is not zero.
- Numerator and Denominator: In a fraction a/b, 'a' is the numerator, and 'b' is the denominator.
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.
