Simplify: 18 / (2√5 - 3√2) - Step-by-Step Solution

Let's dive into the world of mathematics and tackle the simplification of the expression $\frac{18}{2 \sqrt{5}-3 \sqrt{2}}$. This type of problem often appears in algebra and requires a good understanding of radicals and how to manipulate them. This article will walk you through each step, providing clear explanations and insights to help you master similar problems. Our goal is to transform this seemingly complex fraction into its simplest form, making it easier to understand and work with. So, grab your thinking caps, and let's get started!

Understanding the Problem

At its core, this problem is about rationalizing the denominator. The main issue is the presence of radicals (square roots) in the denominator, which makes the expression look complicated. To simplify it, we need to eliminate these radicals from the denominator. The key technique we'll use is multiplying both the numerator and denominator by the conjugate of the denominator. But before we jump into the solution, let’s break down the components of the expression.

• The Numerator: We have a simple whole number, 18. This part is straightforward.
• The Denominator: This is where things get interesting. We have $2 \sqrt{5} - 3 \sqrt{2}$. This is a binomial expression involving square roots. The presence of this binomial in the denominator is what necessitates the rationalization process.
• Radicals: The terms $\sqrt{5}$ and $\sqrt{2}$ are irrational numbers, meaning they cannot be expressed as a simple fraction. Dealing with these requires specific techniques, such as the conjugate method we'll use here.

Understanding these components is the first step in simplifying the expression. Now, let’s move on to the strategy we'll use to solve this problem: rationalizing the denominator.

The Strategy: Rationalizing the Denominator

The primary goal in simplifying expressions like this is to rationalize the denominator. What does that mean? It means we want to eliminate any square roots (or other radicals) from the denominator. The way we achieve this is by multiplying the denominator (and, of course, the numerator to keep the fraction equivalent) by the conjugate of the denominator. But what exactly is a conjugate?

The conjugate of a binomial expression in the form $a - b$ is $a + b$, and vice versa. So, for our denominator, $2 \sqrt{5} - 3 \sqrt{2}$, the conjugate is $2 \sqrt{5} + 3 \sqrt{2}$. The magic of using conjugates comes from the difference of squares formula: $(a - b)(a + b) = a^2 - b^2$. When we multiply the denominator by its conjugate, the middle terms cancel out, leaving us with an expression without radicals.

Here’s the plan:

1. Identify the conjugate of the denominator.
2. Multiply both the numerator and the denominator by this conjugate.
3. Simplify the resulting expression.

This strategy is a cornerstone technique in algebra for dealing with radicals in fractions. By following these steps, we can transform our original expression into a much simpler form. Now, let's put this strategy into action and perform the calculations.

Step-by-Step Solution

Let’s break down the simplification process into manageable steps. This will make it easier to follow along and understand each action.

Step 1: Identify the Conjugate

The first thing we need to do is identify the conjugate of the denominator, which is $2 \sqrt{5} - 3 \sqrt{2}$. As we discussed earlier, the conjugate is formed by changing the sign between the terms. So, the conjugate of our denominator is:

$2 \sqrt{5} + 3 \sqrt{2}$

This is the expression we'll use to multiply both the numerator and the denominator.

Step 2: Multiply by the Conjugate

Now, we multiply both the numerator and the denominator of our original expression by the conjugate we just found:

$\frac{18}{2 \sqrt{5}-3 \sqrt{2}} \times \frac{2 \sqrt{5}+3 \sqrt{2}}{2 \sqrt{5}+3 \sqrt{2}}$

This might look a bit daunting, but remember, we're just multiplying fractions. We multiply the numerators together and the denominators together:

Numerator: $18 \times (2 \sqrt{5} + 3 \sqrt{2})$

Denominator: $(2 \sqrt{5} - 3 \sqrt{2}) \times (2 \sqrt{5} + 3 \sqrt{2})$

Step 3: Simplify the Numerator

Let's start by simplifying the numerator. We distribute the 18 across the terms inside the parentheses:

$18 \times (2 \sqrt{5} + 3 \sqrt{2}) = 18 \times 2 \sqrt{5} + 18 \times 3 \sqrt{2} = 36 \sqrt{5} + 54 \sqrt{2}$

So, the simplified numerator is $36 \sqrt{5} + 54 \sqrt{2}$.

Step 4: Simplify the Denominator

Now, let's tackle the denominator. This is where the magic of the conjugate comes into play. We're multiplying $(2 \sqrt{5} - 3 \sqrt{2})$ by its conjugate $(2 \sqrt{5} + 3 \sqrt{2})$. This is in the form of $(a - b)(a + b)$, which, as we know, equals $a^2 - b^2$. Let's apply this:

$(2 \sqrt{5} - 3 \sqrt{2}) \times (2 \sqrt{5} + 3 \sqrt{2}) = (2 \sqrt{5})^2 - (3 \sqrt{2})^2$

Now, we square each term:

$(2 \sqrt{5})^2 = 2^2 \times (\sqrt{5})^2 = 4 \times 5 = 20$

$(3 \sqrt{2})^2 = 3^2 \times (\sqrt{2})^2 = 9 \times 2 = 18$

So, our denominator simplifies to:

$20 - 18 = 2$

The denominator is now a simple whole number, which is exactly what we wanted!

Step 5: Combine and Simplify the Fraction

Now that we've simplified both the numerator and the denominator, let's put them back together:

$\frac{36 \sqrt{5} + 54 \sqrt{2}}{2}$

We can simplify this fraction further by dividing both terms in the numerator by the denominator:

$\frac{36 \sqrt{5}}{2} + \frac{54 \sqrt{2}}{2} = 18 \sqrt{5} + 27 \sqrt{2}$

And there we have it! Our original expression, after simplification, is $18 \sqrt{5} + 27 \sqrt{2}$.

Final Result

After carefully following each step, we have successfully simplified the expression $\frac{18}{2 \sqrt{5}-3 \sqrt{2}}$. The final simplified form is:

$18 \sqrt{5} + 27 \sqrt{2}$

This is a much cleaner and more manageable form compared to the original expression. By rationalizing the denominator, we've eliminated the radicals from the denominator and made the expression easier to understand and use in further calculations.

Key Takeaways

Throughout this simplification process, we've touched on several important mathematical concepts and techniques. Let's recap the key takeaways to solidify your understanding:

• Rationalizing the Denominator: This is a crucial technique for simplifying expressions with radicals in the denominator. It involves eliminating the radicals to make the expression easier to work with.
• Conjugates: Understanding conjugates is essential for rationalizing denominators. The conjugate of $a - b$ is $a + b$, and vice versa. Multiplying by the conjugate allows us to use the difference of squares formula.
• Difference of Squares: The formula $(a - b)(a + b) = a^2 - b^2$ is a powerful tool for simplifying expressions. It's particularly useful when dealing with conjugates.
• Step-by-Step Approach: Breaking down complex problems into smaller, manageable steps makes the solution process clearer and less intimidating. This approach is valuable for tackling any mathematical challenge.

By mastering these concepts and techniques, you'll be well-equipped to handle similar problems involving radicals and rationalizing denominators.

Practice Problems

To further enhance your understanding and skills, let's look at a couple of practice problems. Working through these will help you solidify the concepts we've covered and build confidence in your ability to simplify similar expressions.

Practice Problem 1

Simplify the expression: $\frac{10}{\sqrt{3} + \sqrt{2}}$

Hint: Identify the conjugate of the denominator and multiply both the numerator and the denominator by it. Remember to simplify your answer as much as possible.

Practice Problem 2

Simplify the expression: $\frac{4}{3 \sqrt{2} - \sqrt{5}}$

Hint: This problem is similar to the one we solved in the article. Pay close attention to the steps involved in rationalizing the denominator and simplifying the resulting expression.

Working through these practice problems will reinforce your understanding of the techniques discussed and help you become more proficient at simplifying expressions with radicals.

Conclusion

Simplifying expressions with radicals, like our example $\frac{18}{2 \sqrt{5}-3 \sqrt{2}}$, might seem daunting at first, but by understanding the underlying principles and following a systematic approach, it becomes quite manageable. The key is to rationalize the denominator, and the conjugate method is your best friend in this endeavor. Remember the importance of conjugates, the difference of squares formula, and breaking down the problem into smaller steps.

We've walked through each step of the solution, from identifying the conjugate to simplifying the final fraction. We've also highlighted key takeaways and provided practice problems to help you master this technique. With consistent practice, you'll become more confident and proficient in simplifying complex mathematical expressions.

Keep exploring the world of mathematics, and remember, every complex problem is just a series of simpler steps waiting to be unraveled! For further learning, you might find helpful resources on websites like Khan Academy's Algebra Section, which offers a wealth of information and practice exercises on algebraic concepts, including radicals and rationalization.