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Last updated on May 26th, 2025

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Square Root of 6500

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If a number is multiplied by itself, the result is a square. The inverse of squaring a number is taking its square root. Square roots are used in various fields such as engineering, finance, and more. Here, we will discuss the square root of 6500.

Square Root of 6500 for Indian Students
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What is the Square Root of 6500?

The square root is the inverse operation of squaring a number. 6500 is not a perfect square. The square root of 6500 can be expressed in both radical and exponential forms. In radical form, it is expressed as √6500, and in exponential form as (6500)^(1/2). √6500 ≈ 80.6226, which is an irrational number because it cannot be represented as a fraction p/q, where p and q are integers and q ≠ 0.

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Finding the Square Root of 6500

For perfect square numbers, the prime factorization method is used. However, for non-perfect square numbers, methods such as long division and approximation are more suitable. Let us explore these methods:

 

  • Prime factorization method
  • Long division method
  • Approximation method
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Square Root of 6500 by Prime Factorization Method

Prime factorization involves breaking down a number into its prime factors. Let's break down 6500 into its prime factors:

 

Step 1: Find the prime factors of 6500. Breaking it down, we get 2 x 2 x 5 x 5 x 13 x 5: 2^2 x 5^3 x 13

 

Step 2: Pair the prime factors. Since 6500 is not a perfect square, the factors cannot be perfectly paired.

 

Thus, calculating √6500 using prime factorization alone is not feasible.

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Square Root of 6500 by Long Division Method

The long division method is particularly useful for non-perfect square numbers. Here’s how to find the square root using this method:

 

Step 1: Group the digits of 6500 from right to left as 65 and 00.

 

Step 2: Find the largest integer n whose square is less than or equal to 65. n is 8 because 8 x 8 = 64.

 

Step 3: Subtract 64 from 65 to get a remainder of 1, and bring down the next pair of zeros to make it 100.

 

Step 4: Double the divisor, which is now 16, and determine a new digit to append to the divisor such that the new divisor times this digit is less than or equal to 100.

 

Step 5: The next digit is 0, so the new divisor is 160, and the new dividend is 100.

 

Step 6: Subtract 160 x 0 from 100 to get a remainder of 100. Bring down the next pair of zeros to get 10000.

 

Step 7: Continue this process to find the square root to the desired decimal places.

 

The square root of 6500 is approximately 80.62.

 

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Square Root of 6500 by Approximation Method

The approximation method is a simpler way to find square roots, especially when a precise value is not necessary. Here's how to apply it for 6500:

 

Step 1: Identify the nearest perfect squares. 6400 (80^2) and 6561 (81^2) are the closest perfect squares around 6500.

 

Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). For example, (6500 - 6400) / (6561 - 6400) = 100 / 161 = 0.6211. Adding this to 80 gives us approximately 80.62.

 

Therefore, the square root of 6500 is approximately 80.62.

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Common Mistakes and How to Avoid Them in the Square Root of 6500

Students often make mistakes when finding square roots, such as neglecting the negative square root or skipping steps in the long division method. Let's discuss some common mistakes in detail.

Mistake 1

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Forgetting about the negative square root

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It’s important to teach students that every positive number has both positive and negative square roots. However, we usually consider only the positive square root in practical applications.

For example, √50 = ±7.07.

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Square Root of 6500 Examples

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Max, the Girl Character from BrightChamps

Problem 1

Can you help Max find the area of a square box if its side length is given as √6500?

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The area of the square is approximately 6500 square units.

Explanation

The area of a square = side^2.

If the side length is √6500, then

Area = (√6500)^2 = 6500.

Therefore, the area is approximately 6500 square units.

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Max, the Girl Character from BrightChamps

Problem 2

A square-shaped plot measuring 6500 square meters is built. If each of the sides is √6500, what is the area of half the plot?

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3250 square meters

Explanation

Since the plot is square-shaped, dividing the area by 2 gives us half the plot's area.

6500 / 2 = 3250 square meters.

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Max, the Girl Character from BrightChamps

Problem 3

Calculate √6500 x 5.

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Approximately 403.113

Explanation

First, find the square root of 6500, which is approximately 80.6226.

Then multiply by 5: 80.6226 x 5 ≈ 403.113.

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Max, the Girl Character from BrightChamps

Problem 4

What will be the square root of (6400 + 100)?

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The square root is 81.

Explanation

First, calculate the sum: 6400 + 100 = 6500.

Then find the square root: √6500 ≈ 80.6226, rounded to the nearest whole number is 81.

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Max, the Girl Character from BrightChamps

Problem 5

Find the perimeter of a rectangle if its length ‘l’ is √6500 units and the width ‘w’ is 50 units.

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The perimeter of the rectangle is approximately 261.2452 units.

Explanation

Perimeter of a rectangle = 2 × (length + width).

Perimeter = 2 × (√6500 + 50) = 2 × (80.6226 + 50) ≈ 2 × 130.6226 ≈ 261.2452 units.

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FAQ on Square Root of 6500

1.What is √6500 in its simplest form?

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2.Mention the factors of 6500.

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3.Calculate the square of 6500.

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4.Is 6500 a prime number?

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5.6500 is divisible by?

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6.How does learning Algebra help students in India make better decisions in daily life?

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7.How can cultural or local activities in India support learning Algebra topics such as Square Root of 6500?

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8.How do technology and digital tools in India support learning Algebra and Square Root of 6500?

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9.Does learning Algebra support future career opportunities for students in India?

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Professor Greenline from BrightChamps

Important Glossaries for the Square Root of 6500

  • Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. Example: √16 = 4 because 4 x 4 = 16.

 

  • Irrational number: An irrational number cannot be expressed as a simple fraction; it has non-repeating, non-terminating decimals.

 

  • Perfect square: A number that is the square of an integer. Example: 36 is a perfect square because it is 6^2.

 

  • Long division method: A technique used to find the square root of a non-perfect square by dividing the number into groups of two digits from right to left and performing systematic division.

 

  • Approximation: A method of finding a number close to the exact square root by identifying nearby perfect squares and using interpolation.
Professor Greenline from BrightChamps

About BrightChamps in India

At BrightChamps, we see algebra as more than just symbols—it opens doors to endless opportunities! Our mission is to help children all over India develop vital math skills, focusing today on the Square Root of 6500 with special attention to understanding square roots—in a way that’s engaging, lively, and easy to follow. Whether your child is calculating the speed of a passing train, keeping scores during a cricket match, or managing pocket money for the latest gadgets, mastering algebra gives them the confidence needed for everyday situations. Our interactive lessons keep learning simple and fun. As kids in India have varied learning styles, we personalize our approach to match each child. From the busy markets of Mumbai to Delhi’s vibrant streets, BrightChamps brings math to life, making it relatable and exciting throughout India. Let’s make square roots a joyful part of every child’s math journey!
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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.

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Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

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