Simplify: $(\sqrt{6}+\sqrt{11})/(\sqrt{5}+\sqrt{3})$ Solution

Hey math enthusiasts! Ever stumbled upon a radical expression that looks like it belongs in a cryptic puzzle? Well, today we're diving headfirst into one of those! We're going to tackle the quotient $(\sqrt{6}+\sqrt{11})/(\sqrt{5}+\sqrt{3})$. This might seem daunting at first, but don't worry, we'll break it down step-by-step, making sure everyone can follow along. Our mission is to simplify this expression and find its true value. So, grab your thinking caps, and let's get started!

Rationalizing the Denominator: The Key to Simplification

The first thing you'll notice when dealing with quotients involving radicals is that pesky radical in the denominator. It's like a tiny pebble in your shoe – annoying and preventing you from moving smoothly. To get rid of it, we employ a technique called rationalizing the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator.

But what's a conjugate, you ask? Great question! The conjugate of a binomial expression like $(\sqrt{5}+\sqrt{3})$ is simply the same expression with the opposite sign between the terms. So, the conjugate of $(\sqrt{5}+\sqrt{3})$ is $(\sqrt{5}-\sqrt{3})$. Think of it as the expression's mathematical twin, but with a slightly different personality.

Now, why do we multiply by the conjugate? Here's the magic: When you multiply a binomial by its conjugate, you eliminate the radical terms. It's like a mathematical sleight of hand! This happens because of the difference of squares identity: $(a + b)(a - b) = a^2 - b^2$.

In our case, multiplying $(\sqrt{5}+\sqrt{3})$ by $(\sqrt{5}-\sqrt{3})$ gives us $(\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2$. Voila! The radicals in the denominator are gone.

So, let's apply this technique to our original quotient. We'll multiply both the numerator and the denominator by $(\sqrt{5}-\sqrt{3})$:

$$\frac{\sqrt{6}+\sqrt{11}}{\sqrt{5}+\sqrt{3}} * \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}-\sqrt{3}}$$

This might look a bit intimidating, but don't fret! We'll tackle it one step at a time.

Expanding the Numerator: A Careful Dance of Radicals

Now that we've multiplied by the conjugate, we need to expand the numerator. This involves carefully applying the distributive property (or the FOIL method, if you prefer). We'll be multiplying each term in the first binomial $(\sqrt{6}+\sqrt{11})$ by each term in the second binomial $(\sqrt{5}-\sqrt{3})$.

Let's break it down:

• $\sqrt{6} * \sqrt{5} = \sqrt{30}$
• $\sqrt{6} * -\sqrt{3} = -\sqrt{18}$
• $\sqrt{11} * \sqrt{5} = \sqrt{55}$
• $\sqrt{11} * -\sqrt{3} = -\sqrt{33}$

So, the expanded numerator becomes: $\sqrt{30} - \sqrt{18} + \sqrt{55} - \sqrt{33}$.

But we're not done yet! We can simplify some of these radicals. Remember that $\sqrt{18}$ can be written as $\sqrt{9 * 2}$, which simplifies to $3\sqrt{2}$. This is like finding hidden treasures within the radicals!

Therefore, our numerator now looks like this: $\sqrt{30} - 3\sqrt{2} + \sqrt{55} - \sqrt{33}$. Much better, right?

Putting It All Together: The Simplified Quotient

We've done the hard work! Now, let's put everything together. Remember that we multiplied both the numerator and the denominator by the conjugate. We've already simplified the denominator to 2, and we've expanded and simplified the numerator.

So, our quotient now looks like this:

$$\frac{\sqrt{30} - 3\sqrt{2} + \sqrt{55} - \sqrt{33}}{2}$$

And that's it! We've successfully simplified the quotient $(\sqrt{6}+\sqrt{11})/(\sqrt{5}+\sqrt{3})$. We've rationalized the denominator, expanded the numerator, and simplified the radicals. It might have seemed like a complicated journey, but we made it!

This final expression represents the quotient in its simplest form. We can't combine the terms in the numerator any further because they involve different radicals. So, we've reached our destination! You've conquered a challenging radical expression. Give yourselves a pat on the back!

Why Does Rationalizing the Denominator Matter?

You might be wondering, why did we go through all this trouble to rationalize the denominator? It's a fair question! While the original expression and the simplified expression are mathematically equivalent, the simplified form is often much easier to work with.

Think of it like this: imagine trying to add fractions if one of the denominators was a complicated radical expression. It would be a nightmare! Rationalizing the denominator makes calculations much smoother and simpler. It's like paving a smooth road for your mathematical journey.

Also, in many contexts, having a rational denominator is considered the standard form for expressing such quotients. It's like following the rules of grammar in writing – it makes your work clearer and easier to understand.

Real-World Applications: Where Do Radicals Show Up?

Radicals might seem like abstract mathematical concepts, but they actually pop up in various real-world applications. From physics to engineering to computer graphics, radicals play a crucial role.

For example, the distance formula, which uses square roots, is used in navigation, mapping, and even video game development. The Pythagorean theorem, which also involves square roots, is fundamental in architecture and construction. Electrical engineers use radicals to calculate impedance in AC circuits.

So, while simplifying radical expressions might seem like a purely academic exercise, it's actually a valuable skill that can be applied in many fields. You're not just learning math; you're learning a tool that can help you understand the world around you!

Practice Makes Perfect: Keep Those Radicals Flowing!

Like any mathematical skill, simplifying radical expressions takes practice. The more you practice, the more comfortable you'll become with the techniques and the faster you'll be able to solve problems.

Try tackling different types of radical expressions, including those with more complicated numerators and denominators. Experiment with different techniques for simplifying radicals, such as factoring out perfect squares or using the properties of radicals.

Don't be afraid to make mistakes! Mistakes are part of the learning process. When you make a mistake, take the time to understand why you made it and how you can avoid making it in the future.

And most importantly, have fun! Math can be challenging, but it can also be incredibly rewarding. When you conquer a difficult problem, you'll feel a sense of accomplishment that's hard to beat.

Conclusion: You've Mastered the Quotient!

Congratulations, guys! You've successfully navigated the world of radical expressions and found the quotient of $(\sqrt{6}+\sqrt{11})/(\sqrt{5}+\sqrt{3})$. You've learned the importance of rationalizing the denominator, the magic of conjugates, and the power of simplifying radicals. You've also seen how radicals show up in real-world applications, making math not just a subject in school, but a tool for understanding the world.

Keep practicing, keep exploring, and keep those mathematical skills sharp! The world of math is vast and fascinating, and there's always something new to discover. Now go forth and conquer more mathematical challenges!