Simplifying Radicals With Exponents A Step By Step Guide

In this comprehensive guide, we will delve into the process of simplifying radical expressions involving exponents. Specifically, we will tackle the expression $\sqrt{\frac{72 x^{16}}{50 x^{36}}}$, providing a step-by-step solution and clarifying the underlying mathematical principles. This detailed explanation will not only help you solve this particular problem but also equip you with the skills to simplify similar expressions confidently. Whether you are a student learning about radicals and exponents or someone looking to refresh your algebra skills, this guide is designed to provide a clear and thorough understanding.

Breaking Down the Problem

Before we dive into the solution, let's understand the key components of the problem. We are given the expression $\sqrt{\frac{72 x^{16}}{50 x^{36}}}$, which involves a square root, a fraction, and variables raised to exponents. To simplify this, we need to address each component systematically. Our main goal is to reduce the expression to its simplest form, where the radicand (the expression inside the square root) has no perfect square factors and the exponents are minimized.

Understanding the Components

1. Square Root: The square root symbol ($\sqrt{}$) indicates that we are looking for a value that, when multiplied by itself, gives the expression inside the root. In other words, we want to find a simplified expression that, when squared, equals $\frac{72 x^{16}}{50 x^{36}}$.
2. Fraction: The fraction $\frac{72 x^{16}}{50 x^{36}}$ represents a ratio between two expressions. Simplifying this fraction involves reducing both the numerical coefficients (72 and 50) and the variable terms ($x^{16}$ and $x^{36}$) to their lowest terms.
3. Exponents: The exponents (16 and 36) indicate the number of times the variable $x$ is multiplied by itself. Simplifying expressions with exponents often involves using the laws of exponents, such as the quotient rule, which states that $\frac{x^a}{x^b} = x^{a-b}$.

Initial Simplification

To begin, let's focus on simplifying the fraction inside the square root. We'll start by reducing the numerical coefficients and then move on to the variable terms. This initial simplification will make the expression easier to work with and set the stage for further simplification.

Step-by-Step Solution

Now, let's walk through the solution step-by-step, ensuring we cover each simplification technique in detail.

Step 1: Simplify the Numerical Coefficients

The first step is to simplify the fraction formed by the numerical coefficients, which is $\frac{72}{50}$. To do this, we need to find the greatest common divisor (GCD) of 72 and 50 and divide both numbers by it. The GCD of 72 and 50 is 2. Dividing both numbers by 2, we get:

$\frac{72}{50} = \frac{72 \div 2}{50 \div 2} = \frac{36}{25}$

This simplification reduces the complexity of the fraction, making it easier to manage within the square root.

Step 2: Simplify the Variable Terms

Next, we simplify the variable terms $\frac{x^{16}}{x^{36}}$. To do this, we use the quotient rule of exponents, which states that when dividing terms with the same base, we subtract the exponents:

$\frac{x^{16}}{x^{36}} = x^{16 - 36} = x^{-20}$

So, the fraction of variable terms simplifies to $x^{-20}$.

Step 3: Combine the Simplified Terms

Now, we combine the simplified numerical coefficients and variable terms to rewrite the expression inside the square root:

$\frac{72 x^{16}}{50 x^{36}} = \frac{36}{25} x^{-20}$

This combined expression is much simpler than the original, and we can now focus on applying the square root.

Step 4: Apply the Square Root

We now apply the square root to the simplified expression:

$\sqrt{\frac{72 x^{16}}{50 x^{36}}} = \sqrt{\frac{36}{25} x^{-20}}$

To simplify the square root of a product, we can take the square root of each factor separately:

$\sqrt{\frac{36}{25} x^{-20}} = \sqrt{\frac{36}{25}} \cdot \sqrt{x^{-20}}$

Step 5: Simplify the Square Root of the Numerical Fraction

We simplify the square root of the numerical fraction:

$\sqrt{\frac{36}{25}} = \frac{\sqrt{36}}{\sqrt{25}} = \frac{6}{5}$

Step 6: Simplify the Square Root of the Variable Term

To simplify the square root of $x^{-20}$, we use the rule $\sqrt{x^a} = x^{\frac{a}{2}}$:

$\sqrt{x^{-20}} = x^{\frac{-20}{2}} = x^{-10}$

Step 7: Combine the Simplified Terms

Now, we combine the simplified numerical and variable terms:

$\frac{6}{5} \cdot x^{-10}$

Step 8: Express with Positive Exponent

To express the variable term with a positive exponent, we use the rule $x^{-a} = \frac{1}{x^a}$:

$x^{-10} = \frac{1}{x^{10}}$

So, the final simplified expression is:

$\frac{6}{5} \cdot \frac{1}{x^{10}} = \frac{6}{5 x^{10}}$

Final Answer

Therefore, the simplified form of $\sqrt{\frac{72 x^{16}}{50 x^{36}}}$ is $\frac{6}{5 x^{10}}$.

Conclusion

In this comprehensive guide, we have successfully simplified the expression $\sqrt{\frac{72 x^{16}}{50 x^{36}}}$ step by step. We began by breaking down the problem into its key components: the square root, the fraction, and the exponents. We then systematically simplified the numerical coefficients and variable terms, applied the square root, and expressed the final answer with a positive exponent. This detailed process not only provides the solution but also enhances your understanding of simplifying radical expressions with exponents.

By following these steps, you can confidently tackle similar problems and strengthen your algebra skills. Remember to focus on simplifying each component individually and then combining them to reach the final simplified form. This methodical approach will help you navigate complex mathematical expressions with ease and accuracy.

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