Simplifying The Quotient Of Radicals: A Step-by-Step Guide

Hey guys! Let's dive into this math problem together. We're going to figure out the quotient of the expression:

$$\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

This might look a little intimidating at first, but don't worry, we'll break it down step by step. Our main goal here is to simplify this expression and get rid of those pesky square roots in the denominator. How do we do that? We use a neat trick called rationalizing the denominator. So, grab your thinking caps, and let's get started!

Understanding the Problem

Okay, before we jump into solving, let's make sure we understand exactly what we're dealing with. The problem asks us to find the quotient of a fraction where both the numerator and the denominator contain square roots. Specifically, we have the fraction:

$$\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

The key here is to simplify this expression. We want to get rid of the square roots in the denominator. Having square roots in the denominator isn't considered simplified form in mathematics. Think of it like having a fraction that isn't in its lowest terms – we always want to simplify it further. So, our mission is clear: rationalize the denominator.

Why do we need to rationalize the denominator, you might ask? Well, it's mostly about convention and making things easier to work with. When we have a rational denominator (a denominator without square roots), it's much simpler to compare fractions, perform further calculations, and generally work with the expression. It's like tidying up your workspace before you start a project – it just makes everything flow smoother.

So, we need to find a way to transform the denominator, $\sqrt{13}-\sqrt{11}$, into a rational number. How do we do that? This is where the concept of the conjugate comes into play. Stay tuned, because we're about to unravel this mystery!

Rationalizing the Denominator: The Conjugate

Alright, let's talk about the magic trick that will help us get rid of those square roots in the denominator: the conjugate. The conjugate is our secret weapon for rationalizing denominators, and it's actually quite simple to understand. For a binomial expression (an expression with two terms) like our denominator, $\sqrt{13}-\sqrt{11}$, the conjugate is formed by simply changing the sign between the terms.

So, what's the conjugate of $\sqrt{13}-\sqrt{11}$? You guessed it! It's $\sqrt{13}+\sqrt{11}$. See? Just flipped the minus sign to a plus sign. Easy peasy!

Now, why is the conjugate so special? Here's where the magic happens. When we multiply an expression by its conjugate, we eliminate the square roots. This is because of a handy algebraic identity: $(a - b)(a + b) = a^2 - b^2$. Notice how the middle terms cancel out, leaving us with just the squares of the terms. This is exactly what we want!

Let's see how this applies to our problem. If we multiply the denominator, $\sqrt{13}-\sqrt{11}$, by its conjugate, $\sqrt{13}+\sqrt{11}$, we get:

$(\sqrt{13}-\sqrt{11})(\sqrt{13}+\sqrt{11}) = (\sqrt{13})^2 - (\sqrt{11})^2 = 13 - 11 = 2$

Boom! The square roots are gone! We've successfully rationalized the denominator... well, at least we've taken the first step. Remember, to keep the fraction equivalent, we need to multiply both the numerator and the denominator by the conjugate. We can't just multiply the denominator and call it a day. It's like adding something to one side of an equation – you have to do it to the other side too to maintain balance. So, let's see how this looks when we apply it to the whole fraction.

Applying the Conjugate to the Fraction

Okay, guys, we've got the conjugate, we know why it works, now let's put it into action! Remember, our original fraction is:

$$\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

And we know the conjugate of the denominator is $\sqrt{13}+\sqrt{11}$. To rationalize the denominator, we need to multiply both the numerator and the denominator by this conjugate. This is like multiplying by a fancy form of 1, because any number (except 0) divided by itself is 1. Multiplying by 1 doesn't change the value of the fraction, just its appearance.

So, let's do it! We'll multiply both the top and bottom of the fraction by $\sqrt{13}+\sqrt{11}$:

$$\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}} \cdot \frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}+\sqrt{11}}$$

Now we need to multiply out the numerators and the denominators. Let's start with the numerator. We're multiplying $(\sqrt{13}+\sqrt{11})$ by itself, which is the same as squaring it. We can use the formula $(a + b)^2 = a^2 + 2ab + b^2$ or simply use the good old FOIL (First, Outer, Inner, Last) method:

$(\sqrt{13}+\sqrt{11})(\sqrt{13}+\sqrt{11}) = (\sqrt{13})^2 + 2(\sqrt{13})(\sqrt{11}) + (\sqrt{11})^2 = 13 + 2\sqrt{143} + 11$

So the numerator simplifies to $13 + 2\sqrt{143} + 11$. Now let's tackle the denominator. We already know that multiplying an expression by its conjugate eliminates the square roots. We calculated it earlier, but let's do it again for good measure:

$(\sqrt{13}-\sqrt{11})(\sqrt{13}+\sqrt{11}) = (\sqrt{13})^2 - (\sqrt{11})^2 = 13 - 11 = 2$

The denominator simplifies to 2. Awesome! Now we have a fraction with a rational denominator. Let's put it all together:

$$\frac{13 + 2\sqrt{143} + 11}{2}$$

We're almost there! We just need to simplify the numerator a little bit more.

Simplifying the Result

Alright, we've done the heavy lifting of rationalizing the denominator. Now it's just a matter of tidying things up and presenting our final answer in the simplest form. We've got the fraction:

$$\frac{13 + 2\sqrt{143} + 11}{2}$$

First, let's combine the like terms in the numerator. We have 13 and 11, which are both constants. Adding them together gives us 24. So our fraction now looks like this:

$$\frac{24 + 2\sqrt{143}}{2}$$

Now, take a close look at the numerator. Notice anything? Both terms, 24 and $2\sqrt{143}$, are divisible by 2. This means we can simplify the fraction further by dividing both the numerator and the denominator by 2. It's like reducing a regular fraction to its lowest terms.

Dividing 24 by 2 gives us 12. Dividing $2\sqrt{143}$ by 2 gives us $\sqrt{143}$. And dividing the denominator, 2, by 2 gives us 1. So, our simplified fraction is:

$$\frac{12 + \sqrt{143}}{1}$$

And anything divided by 1 is just itself, so our final, simplified answer is:

$$12 + \sqrt{143}$$

We did it! We successfully rationalized the denominator and simplified the expression. Give yourselves a pat on the back!

Final Answer

So, after all that simplifying and rationalizing, we've arrived at our final answer. The quotient of the original expression,

$$\frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}-\sqrt{11}}$$

is:

$$12 + \sqrt{143}$$

And that's it! We took a potentially scary-looking problem and broke it down into manageable steps. We learned about rationalizing the denominator, using the conjugate, and simplifying expressions. Hopefully, you feel a little more confident tackling similar problems in the future. Remember, math is like building with LEGOs – each concept builds upon the previous one. So keep practicing, keep learning, and you'll be amazed at what you can achieve!