# What is the Square Root of 39? A Comprehensive Guide

Unlock the mystery of what is the square root of 39 with our comprehensive guide. Learn the basics, calculation, properties, and applications of square roots.

## Introduction

If you’ve ever wondered what the square root of 39 is, then this article is for you. Square roots are an essential part of mathematics and have practical applications in various fields. In this article, we will explore the basics of square roots, how to calculate the square root of 39, properties of square roots, and their applications. By the end of this article, you will have a better understanding of what square roots are and how they work.

## The Basics of Square Roots

To understand the square root of 39, we must first understand what square roots are. A square root is a number that, when multiplied by itself, gives the original number. In other words, the square root of a number is the value that, when squared, equals that number. The symbol for square root is √. For example, the square root of 25 is 5 because 5 x 5 = 25.

To find the square root of a number, we can use the long division method or a calculator. The long division method involves finding the largest perfect square that is less than the number and subtracting it from the number. We then bring down the next two digits and repeat the process until we have the desired level of accuracy. However, for larger numbers, it is more convenient to use a calculator. Most scientific calculators have a square root function that can be used to find the square root of any number.

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## Square Root of 39

Now that we understand the basics of square roots, let’s calculate the square root of 39. Using a scientific calculator, we find that the square root of 39 is approximately 6.2449979984. However, this is a decimal approximation, and we can simplify it further.

To simplify the square root of 39, we can factor the number 39 into its prime factors, which are 3 and 13. We can then take out the square factors, which in this case is 3. This leaves us with 3√13 as the simplified form of the square root of 39. Therefore, the square root of 39 is 3√13.

## Properties of Square Roots

Square roots have several properties that can be useful in calculations. The three main properties are the product property, quotient property, and power property.

The product property of square roots states that the square root of the product of two numbers is equal to the product of their square roots. In other words, √(ab) = √a x √b. For example, the square root of 12 can be written as √(4 x 3), which is equal to √4 x √3 = 2√3.

The quotient property of square roots states that the square root of the quotient of two numbers is equal to the quotient of their square roots. In other words, √(a/b) = √a / √b. For example, the square root of 20 can be written as √(80 / 4), which is equal to √80 / √4 = 4√5 / 2 = 2√5.

The power property of square roots states that the square root of a number raised to a power is equal to the number raised to half the power. In other words, √(a^n) = a^(n/2). For example, the square root of 16^3 can be written as √(16^2 x 16), which is equal to 16^(2/2) x 16^(1/2) = 16^1 x 4 = 64.

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## Applications of Square Roots

Square roots have various real-life applications, ranging from construction to medicine. One common application is in calculating the distance between two points in a two-dimensional space using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. Therefore, finding the length of one of the sides requires finding the square root of the sum of the squares of the other two sides.

Square roots also have applications in finance, where they are used to calculate compound interest. Compound interest is calculated by taking the initial principal amount and adding the interest earned at the end of each period. The formula for calculating compound interest involves taking the square root of the final amount divided by the initial amount and subtracting 1.

In mathematics, square roots have applications in algebra, calculus, and geometry. They are used to solve equations and to find the roots of polynomial equations. In geometry, square roots are used to find the length of sides, the area of circles, and the volume of spheres.

## Conclusion

Square roots are an essential part of mathematics and have practical applications in various fields. In this article, we explored the basics of square roots and how to calculate the square root of 39. We also discussed the properties of square roots, their applications, and real-life examples. It is important to understand square roots and their calculations because they have practical uses in everyday life and are fundamental to advanced mathematics. With this knowledge, you can better appreciate the significance of square roots and their role in modern society.

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