What is the Square Root of 98/72: A Comprehensive Guide

When tackling mathematical problems, evaluating the square root of a fraction is a valuable skill that can simplify complex equations. In this article, we will explore the step-by-step process of finding the square root of 98/72, along with an in-depth discussion on simplifying fractions and dividing radicals. By following the methods outlined below, you will gain a better understanding of mathematical operations and be able to confidently solve similar problems.

Simplifying the Fraction

Let's begin by simplifying the fraction 98/72. This involves finding the greatest common divisor (GCD) of the numerator and denominator to reduce the fraction to its simplest form.

Step 1: Determine the GCD of 98 and 72

The greatest common divisor of 98 and 72 is 2. Applying this to the fraction, we get:

98 ÷ 2 49 72 ÷ 2 36

Therefore, the simplified fraction is 49/36.

Step 2: Finding the Square Root of the Simplified Fraction

The square root of a fraction can be found by taking the square root of the numerator and the denominator separately.

sqrt{frac{49}{36}} frac{sqrt{49}}{sqrt{36}} frac{7}{6}

Thus, the square root of 98/72 is 7/6.

Alternative Methods for Solving the Square Root of Fractions

There are other methods to solve the square root of a fraction. Let's explore a few additional approaches.

Method 1: Prime Factorization

To solve the square root of 72/98, we can use prime factorization. Decompose the numerator and denominator into their prime factors.

72 2^3 * 3^2 98 2 * 7^2

Next, rewrite the fraction by taking the square root of the numerator and the denominator:

sqrt{frac{2^3 * 3^2}{2 * 7^2}} sqrt{frac{(2^2 * 3^2)}{7^2}} frac{6}{7}

Method 2: Division by Common Integers

A simpler approach is to divide both the numerator and the denominator by any common integers that can be divided without remainder. In our case, we can divide both by 2:

frac{72}{2} 36 frac{98}{2} 49

Now, taking the square root:

sqrt{frac{36}{49}} frac{6}{7}

Additional Tips for Dealing with Square Roots

It is important to understand that the square root of a number squared results in the absolute value of that number. This means that:

sqrt{7^2} 7 sqrt{(-7)^2} 7

Additionally, it is crucial to focus on the units column of the fraction when dealing with square roots, as they can be canceled out to simplify the problem. For example:

sqrt{frac{72}{98}} sqrt{frac{36}{49}} frac{6}{7}

By using these methods, you can effectively simplify and solve square roots of fractions. For further assistance or to explore mathematical problems in greater depth, you can visit reputable websites dedicated to mathematics.