Rationalize Denominator: 7√8 / (√8 - 1) Solution

Hey guys! Today, we're diving into a common algebra problem: rationalizing the denominator. This might sound intimidating, but it's actually a pretty straightforward process. We're going to tackle the expression 7√8 / (√8 - 1). So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into the solution, let's break down what it means to rationalize a denominator. In simple terms, it means getting rid of any square roots (or other radicals) in the bottom part of a fraction. Why do we do this? Well, it's generally considered good mathematical practice to have a simplified expression, and having a rational denominator helps achieve that. Plus, it makes further calculations easier down the road.

Our main goal here is to transform the given expression, 7√8 / (√8 - 1), into an equivalent form where the denominator is a rational number (i.e., no square roots). The key technique we'll use is multiplying both the numerator and the denominator by the conjugate of the denominator. But what exactly is a conjugate? Let's find out!

What is a Conjugate?

The conjugate of a binomial expression (an expression with two terms) like (a - b) is simply (a + b). Similarly, the conjugate of (a + b) is (a - b). Notice the only difference is the sign between the terms. This seemingly small change is incredibly powerful when dealing with square roots because of a neat algebraic trick:

(a - b)(a + b) = a² - b²

This is the difference of squares formula. When we multiply a binomial by its conjugate, the middle terms cancel out, leaving us with the difference of the squares of the two original terms. This is exactly what we need to eliminate square roots!

In our problem, the denominator is (√8 - 1). So, its conjugate is (√8 + 1). We'll be multiplying both the top and bottom of our fraction by this conjugate.

Step-by-Step Solution

Okay, let's get down to the nitty-gritty and solve this problem step-by-step.

Step 1: Identify the Conjugate

As we discussed, the denominator of our expression is (√8 - 1). Therefore, its conjugate is (√8 + 1).

Step 2: Multiply by the Conjugate

Now, we'll multiply both the numerator and the denominator of the expression by the conjugate (√8 + 1). This is crucial because multiplying both the top and bottom by the same value is equivalent to multiplying by 1, which doesn't change the value of the expression, only its form.

So, we have:

(7√8 / (√8 - 1)) * ((√8 + 1) / (√8 + 1))

This looks a bit messy, but don't worry, we'll simplify it in the next step.

Step 3: Distribute and Simplify

Next, we need to distribute the terms in both the numerator and the denominator. Let's start with the numerator:

7√8 * (√8 + 1) = 7√8 * √8 + 7√8 * 1 = 7 * 8 + 7√8 = 56 + 7√8

Now, let's tackle the denominator. Remember our difference of squares formula? This is where it comes in handy:

(√8 - 1)(√8 + 1) = (√8)² - (1)² = 8 - 1 = 7

Notice how the square root disappeared from the denominator! This is exactly what we wanted.

So, our expression now looks like this:

(56 + 7√8) / 7

Step 4: Final Simplification

We're almost there! We can simplify this fraction further by dividing both terms in the numerator by the denominator:

(56 + 7√8) / 7 = 56/7 + (7√8)/7 = 8 + √8

Now, we can simplify √8. Since 8 = 4 * 2, we have √8 = √(4 * 2) = √4 * √2 = 2√2.

Therefore, our final simplified expression is:

8 + 2√2

The Answer

So, after rationalizing the denominator of the expression 7√8 / (√8 - 1), we arrive at the simplified form: 8 + 2√2. Awesome!

Why This Matters

You might be thinking, "Okay, we solved it, but why is this important?" Rationalizing the denominator isn't just a mathematical exercise; it has practical applications. It helps in:

• Simplifying expressions: Makes expressions easier to work with in further calculations.
• Comparing values: Allows for easier comparison of fractions with different denominators.
• Advanced math: It's a fundamental skill needed for calculus and other higher-level math courses.

Common Mistakes to Avoid

When rationalizing denominators, it's easy to make a few common mistakes. Keep an eye out for these:

• Forgetting to multiply both numerator and denominator: Remember, you're essentially multiplying by 1, so you need to apply the conjugate to both parts of the fraction.
• Incorrectly applying the distributive property: Make sure you multiply each term in the numerator and denominator by the appropriate terms.
• Not simplifying completely: Always check if you can simplify the resulting expression further, like we did with √8 in our example.

Practice Makes Perfect

The best way to master rationalizing denominators is to practice! Try working through similar problems. Here's a quick practice problem for you guys:

Rationalize the denominator of the expression: 3 / (2 + √5)

(Hint: The conjugate of (2 + √5) is (2 - √5))

Conclusion

Rationalizing the denominator might seem tricky at first, but with a little practice, you'll become a pro! Remember the key steps: identify the conjugate, multiply, simplify, and watch out for common mistakes. By mastering this technique, you'll not only improve your algebra skills but also gain a deeper understanding of mathematical principles. Keep practicing, and you'll be solving these problems in no time! Good luck, and happy math-ing!