Simplify The Radical Expression: A Step-by-Step Guide

Hey guys! Today, we're going to break down a radical expression and simplify it step by step. This is a common type of problem you'll see in algebra, and mastering it will definitely boost your math skills. So, let's jump right into it!

Understanding the Problem

Before we dive into the solution, let's make sure we understand the problem. We are given the expression $\sqrt{\frac{20xy}{5x^3y^3}}$ and we need to simplify it. This involves getting rid of the radical in the denominator and making the expression as clean as possible. Remember, the key here is to know your exponent rules and how to handle square roots.

Breaking Down the Expression

Our starting point is the expression under the square root: $\frac{20xy}{5x^3y^3}$.

First, let's simplify the fraction by dividing the coefficients and using exponent rules. When dividing terms with the same base, we subtract the exponents.

$\frac{20}{5} = 4$

$\frac{x}{x^3} = x^{1-3} = x^{-2}$

$\frac{y}{y^3} = y^{1-3} = y^{-2}$

So, our expression becomes $4x^{-2}y^{-2}$.

Rewriting with Positive Exponents

To make things easier to work with, let's rewrite the expression with positive exponents. Remember that $a^{-n} = \frac{1}{a^n}$.

$4x^{-2}y^{-2} = \frac{4}{x^2y^2}$

Applying the Square Root

Now, we need to apply the square root to the simplified fraction:

$\sqrt{\frac{4}{x^2y^2}}$

We can take the square root of the numerator and the denominator separately:

$\frac{\sqrt{4}}{\sqrt{x^2y^2}}$

$\sqrt{4} = 2$

$\sqrt{x^2y^2} = xy$ (since $x > 0$ and $y > 0$)

So, the simplified expression is $\frac{2}{xy}$.

Step-by-Step Solution

Here’s a detailed breakdown of how we arrive at the simplified expression. This covers all the little steps to ensure clarity and understanding. Remember, practice makes perfect, so feel free to try similar problems on your own.

Step 1: Simplify the Fraction Inside the Square Root

We begin with the expression $\sqrt{\frac{20xy}{5x^3y^3}}$. Our first task is to simplify the fraction inside the square root. We divide the coefficients and simplify the variables using exponent rules. Simplify $\frac{20xy}{5x^3y^3}$ as follows:

• Divide the coefficients: $\frac{20}{5} = 4$.
• Simplify the $x$ terms: $\frac{x}{x^3} = x^{1-3} = x^{-2}$.
• Simplify the $y$ terms: $\frac{y}{y^3} = y^{1-3} = y^{-2}$.

So, the expression inside the square root becomes $4x^{-2}y^{-2}$. Therefore,

$\sqrt{\frac{20xy}{5x^3y^3}} = \sqrt{4x^{-2}y^{-2}}$

Step 2: Rewrite with Positive Exponents

Next, we rewrite the expression $4x^{-2}y^{-2}$ using positive exponents. Recall that $a^{-n} = \frac{1}{a^n}$. Applying this rule, we get:

$4x^{-2}y^{-2} = \frac{4}{x^2y^2}$

So our expression now looks like this:

$\sqrt{4x^{-2}y^{-2}} = \sqrt{\frac{4}{x^2y^2}}$

Step 3: Apply the Square Root

Now, we apply the square root to both the numerator and the denominator of the fraction. This gives us:

$\sqrt{\frac{4}{x^2y^2}} = \frac{\sqrt{4}}{\sqrt{x^2y^2}}$

Step 4: Simplify the Square Roots

We simplify the square roots in the numerator and the denominator. We know that $\sqrt{4} = 2$. Also, since $x > 0$ and $y > 0$, we have $\sqrt{x^2y^2} = xy$. Thus, our expression simplifies to:

$\frac{\sqrt{4}}{\sqrt{x^2y^2}} = \frac{2}{xy}$

So, the final simplified expression is $\frac{2}{xy}$.

Common Mistakes to Avoid

When simplifying radical expressions, there are a few common pitfalls to watch out for. Avoiding these mistakes will help you get to the correct answer more consistently.

Forgetting to Simplify Inside the Radical First

One common mistake is trying to take the square root before simplifying the expression inside the radical. Always simplify the fraction or expression inside the square root first. This makes the problem much easier to handle.

Incorrectly Applying Exponent Rules

Make sure you understand and correctly apply the exponent rules. For example, when dividing terms with the same base, you subtract the exponents, not divide them. Similarly, remember that a negative exponent means you take the reciprocal of the base.

Not Simplifying Completely

Ensure that you simplify the expression completely. This means making sure there are no radicals in the denominator and that all fractions are reduced to their simplest form. Double-check your work to catch any remaining simplifications.

Ignoring the Domain

Pay attention to the domain of the variables. In this problem, we are given that $x > 0$ and $y > 0$. This allows us to simplify $\sqrt{x^2y^2}$ to $xy$ without needing absolute value signs. If the domain were not specified, we would need to be more careful.

Practice Problems

To solidify your understanding, here are a few practice problems. Try solving them on your own, and then check your answers. Remember, the more you practice, the better you'll get!

1. Simplify $\sqrt{\frac{18a^3b}{2ab^5}}$ where $a > 0$ and $b > 0$.
2. Simplify $\sqrt{\frac{25x^4y^2}{9z^6}}$ where $x > 0$, $y > 0$, and $z > 0$.
3. Simplify $\sqrt{\frac{32m^5n^3}{8mn^7}}$ where $m > 0$ and $n > 0$.

Conclusion

Simplifying radical expressions can seem tricky at first, but with a clear understanding of the steps and a bit of practice, you'll be simplifying like a pro in no time. Remember to always simplify inside the radical first, apply exponent rules correctly, and ensure your final answer is completely simplified. Keep practicing, and you'll master these types of problems with ease!