Solving Radical Expressions: Finding The Equivalent In Terms Of K

Alright, guys, let's dive into this math problem! We're given the value of K and asked to find the equivalent of another expression in terms of K. This is a classic algebra problem that involves simplifying radical expressions and manipulating them to find relationships. Don't worry; we'll break it down step by step so it's super easy to follow. This problem is all about clever manipulation and understanding how to rationalize denominators. If you're struggling with radicals, don't sweat it; it just takes practice! Let's get started, shall we?

Understanding the Problem and the Given Information

First things first, let's make sure we understand what we're dealing with. We're given:

• K = (√5 - 2) / (√7 + √3)

And we need to find the equivalent of:

• (√5 + 2) / (√7 - √3)

This is a great opportunity to practice our skills in simplifying radical expressions. The core idea is to use the given value of K to rewrite the target expression. This often involves rationalizing the denominators and looking for opportunities to create similar terms or factors. This will help us relate the target expression to K.

Think of it like this: we have a key (K) that can unlock the solution to another expression. Our goal is to find out how this key fits into the other expression. Before we get down to business, remember that rationalizing the denominator means removing any radicals from the bottom of a fraction. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of (a + b) is (a - b), and vice versa. Let's get our hands dirty.

Rationalizing the Denominator of K

Let's start by looking at K = (√5 - 2) / (√7 + √3). Notice how the denominator has radicals. We are going to rationalize the denominator to simplify the expression, which will make our calculations easier later. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator, which is (√7 - √3).

So, K = (√5 - 2) / (√7 + √3) * (√7 - √3) / (√7 - √3).

This gives us:

K = ((√5 - 2)(√7 - √3)) / ((√7 + √3)(√7 - √3)).

Let's simplify the numerator and the denominator separately. The numerator becomes:

(√5 * √7) - (√5 * √3) - (2 * √7) + (2 * √3) = √35 - √15 - 2√7 + 2√3

The denominator becomes:

(√7 * √7) - (√7 * √3) + (√3 * √7) - (√3 * √3) = 7 - 3 = 4.

Therefore, K = (√35 - √15 - 2√7 + 2√3) / 4.

Now, the original goal was not to simplify K, but we rationalized the denominator for a different reason. We want to relate the value of K to the expression we need to find. So, it is safe to assume that the simplification we did is correct.

Working with the Target Expression

Now, let's look at the expression we need to find the equivalent of: (√5 + 2) / (√7 - √3). We'll rationalize its denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is (√7 + √3).

So, ((√5 + 2) / (√7 - √3)) * ((√7 + √3) / (√7 + √3)).

This gives us:

((√5 + 2)(√7 + √3)) / ((√7 - √3)(√7 + √3)).

Let's simplify the numerator and denominator separately. The numerator becomes:

(√5 * √7) + (√5 * √3) + (2 * √7) + (2 * √3) = √35 + √15 + 2√7 + 2√3.

The denominator becomes:

(√7 * √7) + (√7 * √3) - (√3 * √7) - (√3 * √3) = 7 - 3 = 4.

So, our target expression is (√35 + √15 + 2√7 + 2√3) / 4.

Finding the Relationship

Now comes the clever part. We have K = (√35 - √15 - 2√7 + 2√3) / 4 and our target expression (√35 + √15 + 2√7 + 2√3) / 4. Observe the numerators of both expressions.

K’s numerator: √35 - √15 - 2√7 + 2√3

Target expression's numerator: √35 + √15 + 2√7 + 2√3

Notice the key differences. The signs for the √15 and 2√7 terms are different. It looks like if we flip those signs we might find a relationship with K.

Let's try to relate the target expression's numerator to K. It is hard to do directly. We have to manipulate the given K to find a similar structure. Now, the goal is to manipulate K in a way that helps us relate it to our target expression. Multiply both sides of K = (√35 - √15 - 2√7 + 2√3) / 4 by -1. We obtain

-K = (-√35 + √15 + 2√7 - 2√3) / 4.

Add 2√35/4 to both sides. We get

-K + 2√35/4= (√35 + √15 + 2√7 - 2√3) / 4

Now, we are closer to the original expression. But the last two terms are still different. We can keep multiplying by -1. We will start from the beginning.

Notice that if we multiply the numerator of K by -1, we can get something similar to the target expression. First, multiply K by -1, then multiply by -1 again. If we do so, we will get:

• K = (-√35 + √15 + 2√7 - 2√3) / 4.

• K = -1 * (√35 - √15 - 2√7 + 2√3) / 4.

Let's try to modify K to get something similar to (√35 + √15 + 2√7 + 2√3) / 4. Notice that we have a denominator of 4 in both expressions. Notice that the target expression and K have almost the same terms, but there are different signs.

The Final Calculation

Let's go back to the original expressions:

• K = (√5 - 2) / (√7 + √3)
• Target: (√5 + 2) / (√7 - √3)

We can manipulate the target expression: (√5 + 2) / (√7 - √3) = (√5 + 2) / (√7 - √3) * (√7 + √3) / (√7 + √3) = ((√5 + 2)(√7 + √3)) / (7 - 3) = (√35 + √15 + 2√7 + 2√3) / 4.

Now let's manipulate K:

K = (√5 - 2) / (√7 + √3). If we multiply the numerator and denominator by -1: -K = (2 - √5) / (√7 + √3). Multiply both sides by -1 again, the expression would be the same, but not helpful.

Instead, let's try this: 1/K = (√7 + √3) / (√5 - 2).

Rationalize the denominator: 1/K = (√7 + √3) / (√5 - 2) * (√5 + 2) / (√5 + 2) = (√35 + 2√7 + √15 + 2√3) / (5 - 4) = (√35 + 2√7 + √15 + 2√3).

Therefore, we can rewrite the target expression as follows: (√35 + √15 + 2√7 + 2√3) / 4 = (√35 + 2√7 + √15 + 2√3) / 4 = (1/K) / 4 = 4 / K

Let's examine the original expressions.

K = (√5 - 2) / (√7 + √3)

Target Expression = (√5 + 2) / (√7 - √3)

Multiply the numerator and denominator by the conjugate of the denominator.

Target Expression = (√5 + 2) / (√7 - √3) * (√7 + √3) / (√7 + √3) = (√35 + √15 + 2√7 + 2√3) / (7 - 3) = (√35 + √15 + 2√7 + 2√3) / 4.

1 / K = (√7 + √3) / (√5 - 2)

Multiply by conjugate.

1 / K = (√7 + √3) / (√5 - 2) * (√5 + 2) / (√5 + 2) = (√35 + 2√7 + √15 + 2√3) / (5 - 4) = √35 + 2√7 + √15 + 2√3.

Target Expression / (1 / K) = (√35 + √15 + 2√7 + 2√3) / 4 / (√35 + 2√7 + √15 + 2√3) = 1 / 4.

Therefore, Target Expression = 4 / K.

So the correct answer is E) 4/K.

Conclusion

There you have it, guys! We've successfully found the equivalent of the target expression in terms of K. This was a great exercise in algebraic manipulation, and we got to review important techniques like rationalizing denominators and recognizing patterns. Keep practicing, and these types of problems will become second nature. If you ever get stuck on a similar problem, remember to break it down step by step and look for ways to relate the given information to the expression you're trying to find. Keep up the great work, and happy calculating!