145 LearnersLast updated on May 26th, 2025
Square Root of 69.25
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 various fields such as vehicle design, finance, etc. Here, we will discuss the square root of 69.25.
What is the Square Root of 69.25?
The square root is the inverse of the square of the number. 69.25 is not a perfect square. The square root of 69.25 is expressed in both radical and exponential form.
In the radical form, it is expressed as √69.25, whereas (69.25)(1/2) in the exponential form. √69.25 ≈ 8.320, 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 69.25
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers like 69.25, methods such as the long-division method and approximation method are used. Let us now learn the following methods:
1. Prime factorization method
2. Long division method
3. Approximation method
Square Root of 69.25 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Since 69.25 is not a perfect square, prime factorization is not a suitable method.
Instead, we use approximation or long division methods to find the square root.
Square Root of 69.25 by Long Division Method
The long division method is particularly used for non-perfect square numbers. Let us learn how to find the square root using the long division method, step by step:
Step 1: Group the numbers from right to left. In the case of 69.25, consider it as 69 and 25 for the decimal part.
Step 2: Find n whose square is less than or equal to 69. We find n as 8 because 8 × 8 = 64, which is less than 69. The quotient is 8, and the remainder is 69 - 64 = 5.
Step 3: Bring down 25 (from the decimal part), making it 525. Double the quotient (8) to get 16, which will be part of the new divisor.
Step 4: Find n such that 16n × n ≤ 525. Let n be 3. Then, 163 × 3 = 489.
Step 5: Subtract 489 from 525 to get the remainder 36.
Step 6: Since the dividend is less than the divisor, continue with a decimal point and add zeroes to the dividend for precision.
Step 7: The continued steps give the square root of 69.25 as approximately 8.320.
Square Root of 69.25 by Approximation Method
The approximation method is an easy way to find square roots. Let's learn how to find the square root of 69.25 using this method:
Step 1: Identify the closest perfect squares. The nearest perfect squares for 69.25 are 64 and 81. √69.25 falls between 8 and 9.
Step 2: Apply the formula (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square).
Using the formula: (69.25 - 64) ÷ (81 - 64) = 5.25 ÷ 17 ≈ 0.309. Adding this decimal to the smaller whole number: 8 + 0.309 = 8.309,
so the square root of 69.25 is approximately 8.320.
Common Mistakes and How to Avoid Them in the Square Root of 69.25
Students often make mistakes while finding square roots, such as forgetting about the negative square root or skipping steps in the long division method. Here are a few common mistakes and how to avoid them:
Mistake 1
Forgetting about the negative square root
It is important to make students aware that a number has both positive and negative square roots. However, we usually consider only the positive square root due to its practical applications. For example, √50 ≈ 7.07, but there is also -7.07 which should not be overlooked.
Square Root of 69.25 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √90.25?
The area of the square is 90.25 square units.
Explanation
The area of the square = side².
The side length is given as √90.25.
Area of the square = side² = √90.25 × √90.25 = 9.5 × 9.5 = 90.25
Therefore, the area of the square box is 90.25 square units.
Problem 2
A square-shaped building measuring 69.25 square feet is built; if each of the sides is √69.25, what will be the square feet of half of the building?
34.625 square feet
Explanation
To find half of the building's area, divide the total area by 2.
Dividing 69.25 by 2 = 34.625
So half of the building measures 34.625 square feet.
Problem 3
Calculate √69.25 × 5.
41.6
Explanation
First, find the square root of 69.25, which is approximately 8.320. Then, multiply 8.320 by 5. So, 8.320 × 5 = 41.6
Problem 4
What will be the square root of (69.25 + 10.75)?
The square root is approximately 9.
Explanation
To find the square root, first find the sum of (69.25 + 10.75). 69.25 + 10.75 = 80, and then √80 ≈ 8.944.
Therefore, the square root of (69.25 + 10.75) is approximately ±8.944.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √69.25 units and the width ‘w’ is 15 units.
The perimeter of the rectangle is approximately 46.64 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√69.25 + 15) = 2 × (8.320 + 15) ≈ 2 × 23.32 = 46.64 units.
FAQ on Square Root of 69.25
1.What is √69.25 in its simplest form?
2.Is 69.25 a perfect square?
3.Calculate the square of 69.25.
4.Is 69.25 a prime number?
5.What numbers are closest to √69.25?
Important Glossaries for the Square Root of 69.25
- Square root: The square root of a number is the value that, when multiplied by itself, gives the original number. Example: √16 = 4.
- Irrational number: A number that cannot be expressed as a simple fraction (p/q), where p and q are integers and q ≠ 0.
- Long division method: A technique used to find the square root of non-perfect squares by systematically dividing the number.
- Radical form: A way to express the square root of a number using the radical symbol (√). Example: √69.25.
- Approximation method: A method to estimate the square root of a number when it is not a perfect square.
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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.
