216 LearnersLast updated on May 26th, 2025
Square Root of 69
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.
What is the Square Root of 69?
The square root is the inverse of squaring a number. 69 is not a perfect square. The square root of 69 is expressed in both radical and exponential forms.
In radical form, it is expressed as √69, whereas in exponential form, it is written as 691/2.
√69 ≈ 8.30662, 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
The prime factorization method is typically used for perfect square numbers. However, for non-perfect square numbers, we use methods like long division and approximation. Let us now learn the following methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 69 by Prime Factorization Method
The product of prime factors gives the prime factorization of a number. Let's see how 69 is broken down into its prime factors:
Step 1: Finding the prime factors of 69 Breaking it down, we get 3 x 23. Therefore, the prime factorization of 69 is 3¹ x 23¹.
Step 2: Since 69 is not a perfect square, the prime factors cannot be paired.
Therefore, calculating the square root of 69 using prime factorization directly is not feasible.
Square Root of 69 by Long Division Method
The long division method is particularly useful for non-perfect square numbers. In this method, we check for the closest perfect square number to the given number. Here is how to find the square root using the long division method, step by step:
Step 1: To begin with, we need to group the digits of 69 starting from the right. In this case, we group it as 69.
Step 2: Find a number whose square is less than or equal to the leftmost group. Since 8² = 64 is less than 69, the initial quotient is 8.
Step 3: Subtract 64 from 69, resulting in a remainder of 5. Step 4: Bring down pairs of zeros one at a time. The new dividend is 500.
Step 5: Double the quotient and use it as part of the new divisor. In this case, 2 × 8 = 16.
Step 6: Find a digit n such that (160 + n) × n is less than or equal to 500. Here, n is 3 since 163 × 3 = 489.
Step 7: Subtract 489 from 500 to get a remainder of 11.
Step 8: Repeat the steps until the decimal reaches the desired precision.
The approximate value of √69 is 8.306.
Square Root of 69 by Approximation Method
The approximation method is another simple way to find the square roots of a number. Here's how to find the square root of 69 using the approximation method:
Step 1: Find the closest perfect squares around 69. The smaller perfect square is 64 and the larger is 81. √69 falls between √64 = 8 and √81 = 9.
Step 2: Use the formula: ( (Given number - smaller perfect square) / (larger perfect \square - smaller perfect square) ).
Applying the formula: (69 - 64) / (81 - 64) = 5 / 17 ≈ 0.294.
Add this decimal to 8, the square root of 64.
Thus, 8 + 0.294 ≈ 8.294.
Therefore, √69 is approximately 8.294.
Common Mistakes and How to Avoid Them in the Square Root of 69
Students often make mistakes while finding square roots, such as forgetting about the negative square root or skipping steps in long division methods. Let's explore some common mistakes in detail.
Mistake 1
Forgetting about the negative square root
It is important to note that a number has both positive and negative square roots. However, we typically only consider the positive square root in practical applications. For example: √49 = 7, but there is also -7, which should not be ignored.
Mistake 2
Not adding square root symbols in proper places
Incorrect use of the square root symbol is a common mistake. It is essential to simplify numbers inside the square root before proceeding. For example: √(16 + 9) = √25 = 5, not √16 + √9 = 4 + 3 = 7, which is incorrect.
Mistake 3
Not finding the correct value of a non-perfect square
Students must simplify numbers inside the square root for accurate approximation. For example: √50 ≠ 7; the correct answer is √50 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
It is crucial to distinguish between square root and cube root symbols. For example, √8 is different from ∛8.
Mistake 5
Making mistakes in long division
Finding a square root using the long division method involves many steps. Students may skip a step accidentally, leading to an incorrect answer. This can happen during subtraction or elsewhere in the process.
Square root of 69 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √69?
The area of the square is approximately 69 square units.
Explanation
The area of the square = side².
The side length is given as √69.
Area of the square = side² = √69 × √69 = 69.
Therefore, the area of the square box is approximately 69 square units.
Problem 2
A square-shaped garden measures 69 square meters; if each of the sides is √69, what will be the area of half of the garden?
34.5 square meters
Explanation
Since the garden is square-shaped, we can divide the total area by 2 to find half of it.
Dividing 69 by 2, we get 34.5.
So half of the garden measures 34.5 square meters.
Problem 3
Calculate √69 × 3.
Approximately 24.91986
Explanation
First, find the square root of 69, which is approximately 8.30662.
Then, multiply 8.30662 by 3. So, 8.30662 × 3 ≈ 24.91986.
Problem 4
What will be the square root of (49 + 20)?
The square root is approximately 9.11043
Explanation
To find the square root, we need to find the sum of (49 + 20). 49 + 20 = 69, and then √69 ≈ 8.30662.
Therefore, the square root of (49 + 20) is approximately ±8.30662.
Problem 5
Find the perimeter of a rectangle if its length ‘l’ is √69 units and the width ‘w’ is 20 units.
The perimeter of the rectangle is approximately 57.61324 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√69 + 20) = 2 × (8.30662 + 20) ≈ 2 × 28.30662 ≈ 57.61324 units.
FAQ on Square Root of 69
1.What is √69 in its simplest form?
2.Mention the factors of 69.
3.Calculate the square of 69.
4.Is 69 a prime number?
5.69 is divisible by?
6.How does learning Algebra help students in Australia make better decisions in daily life?
7.How can cultural or local activities in Australia support learning Algebra topics such as Square Root of 69?
8.How do technology and digital tools in Australia support learning Algebra and Square Root of 69?
9.Does learning Algebra support future career opportunities for students in Australia?
Important Glossaries for the Square Root of 69
- Square root: A square root is a value that, when multiplied by itself, gives the original number. For example, √9 = 3, since 3 × 3 = 9.
- Irrational number: An irrational number cannot be expressed as a simple fraction, such as π or √69.
- Radical form: A way of expressing roots, such as square or cube roots, using the radical symbol (√).
- Approximation: The process of finding a value that is close enough to the correct answer, typically used for non-perfect squares.
- Perfect square: A number that is the square of an integer, such as 64 (8 × 8) or 81 (9 × 9).
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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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