Simplifying 1/(3+√2): A Step-by-Step Guide

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Simplifying 1/(3+√2): A Step-by-Step Guide

Hey guys! Today, we're diving into a common yet crucial topic in mathematics: simplifying expressions with radicals in the denominator. Specifically, we're going to tackle the expression 1/(3 + √2). You might be wondering, why do we even need to simplify this? Well, in mathematics, it's generally considered good practice to remove radicals from the denominator. This makes expressions easier to work with and compare. So, let's get started and break down how to simplify this expression like pros!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We have a fraction where the denominator is (3 + √2). The issue here is the √2, which is an irrational number. Having an irrational number in the denominator can make further calculations and comparisons difficult. Our goal is to get rid of this radical in the denominator, and we'll do this using a clever trick called "rationalizing the denominator."

Rationalizing the denominator basically means we want to transform the fraction so that the denominator becomes a rational number (a number that can be expressed as a fraction of two integers). Think of it like cleaning up the fraction to make it look nicer and be more manageable. We achieve this by multiplying both the numerator and the denominator by a special term called the conjugate.

What is a Conjugate?

The conjugate of a binomial expression (an expression with two terms) in the form of (a + b) is (a - b). Similarly, the conjugate of (a - b) is (a + b). Notice that the only difference is the sign between the terms. This seemingly simple change is the key to eliminating the radical. In our case, the denominator is (3 + √2), so its conjugate is (3 - √2). Remember this – the conjugate is our secret weapon in this simplification mission!

The Step-by-Step Solution

Now that we understand the problem and the concept of conjugates, let's walk through the steps to simplify 1/(3 + √2):

Step 1: Identify the Conjugate

As we discussed, the conjugate of the denominator (3 + √2) is (3 - √2). Keep this in mind, as we'll be using it in the next step.

Step 2: Multiply by the Conjugate

This is where the magic happens! We're going to multiply both the numerator and the denominator of our original fraction by the conjugate we just identified. This is crucial because multiplying both the top and bottom of a fraction by the same value doesn't change the fraction's overall value – it's like multiplying by 1. So, we have:

(1 / (3 + √2)) * ((3 - √2) / (3 - √2))

Step 3: Distribute and Simplify

Now, we need to multiply out the numerators and the denominators. Let's start with the numerator:

  • 1 * (3 - √2) = 3 - √2

The numerator is straightforward. Now, let's tackle the denominator. This is where the conjugate's special property comes into play. We'll use the distributive property (also known as FOIL – First, Outer, Inner, Last) to multiply the binomials:

  • (3 + √2) * (3 - √2) = (3 * 3) + (3 * -√2) + (√2 * 3) + (√2 * -√2)
  • = 9 - 3√2 + 3√2 - 2

Notice anything interesting? The terms -3√2 and +3√2 cancel each other out! This is the power of the conjugate – it eliminates the radical term when multiplied. So, we're left with:

  • 9 - 2 = 7

Step 4: Write the Simplified Fraction

Now we have our simplified numerator and denominator. Let's put them together to form our final answer:

(3 - √2) / 7

And there you have it! We've successfully simplified the expression 1/(3 + √2) to (3 - √2) / 7. The denominator is now a rational number, and we've eliminated the radical.

Why This Works: The Difference of Squares

You might be wondering why multiplying by the conjugate works so well. The secret lies in a mathematical concept called the "difference of squares." The difference of squares formula states:

(a + b)(a - b) = a² - b²

Notice how the middle terms (the "outer" and "inner" terms in FOIL) cancel out, leaving only the squares of a and b. In our example, a is 3 and b is √2. When we square √2, we get 2, which is a rational number! This is why the conjugate is the perfect tool for rationalizing denominators.

By multiplying (3 + √2) by its conjugate (3 - √2), we're essentially applying the difference of squares pattern. This transforms the denominator into a rational number, achieving our goal.

Common Mistakes to Avoid

When simplifying expressions like this, there are a few common mistakes that students often make. Let's take a look at some of these so you can steer clear of them:

  • Forgetting to multiply both the numerator and denominator: Remember, you need to multiply both the top and bottom of the fraction by the conjugate. If you only multiply the denominator, you're changing the value of the expression.
  • Incorrectly identifying the conjugate: Make sure you only change the sign between the terms. The conjugate of (3 + √2) is (3 - √2), not (-3 - √2) or (3 + √2).
  • Making errors in distribution: Be careful when multiplying the binomials. Use FOIL or another method to ensure you multiply each term correctly.
  • Skipping steps: It's tempting to try and do the simplification in your head, but it's best to write out each step to avoid mistakes. This is especially important when you're first learning this concept.

Practice Makes Perfect

The best way to master simplifying expressions with radicals is to practice! Try working through some similar problems on your own. Here are a few examples you can try:

  • Simplify: 1 / (2 - √3)
  • Simplify: (√5 + 1) / (√5 - 1)
  • Simplify: 4 / (1 + √2)

Remember to follow the same steps: identify the conjugate, multiply both the numerator and denominator, simplify, and write the final answer. If you get stuck, review the steps we outlined earlier or ask for help from a teacher or tutor.

Real-World Applications

You might be wondering, where does this stuff actually get used in the real world? While you might not be simplifying radical expressions in your everyday life, the underlying concepts are crucial in many fields, including:

  • Engineering: Engineers often work with complex equations that involve radicals. Simplifying these expressions is essential for calculations and design.
  • Physics: Physics problems frequently involve square roots and other radicals. Simplifying these expressions can make problems easier to solve.
  • Computer Graphics: Radicals are used in calculations related to distance, lighting, and other graphical elements. Simplifying these expressions can improve performance.
  • Advanced Mathematics: Simplifying radical expressions is a fundamental skill that's used in many higher-level math courses, such as calculus and linear algebra.

So, even though it might seem like a purely mathematical exercise, simplifying radical expressions is a valuable skill that can be applied in a variety of fields.

Conclusion

Alright guys, we've covered a lot in this guide! We've learned how to simplify expressions with radicals in the denominator by rationalizing the denominator. We walked through a step-by-step solution, discussed the concept of conjugates and the difference of squares, and highlighted common mistakes to avoid. We also touched on the real-world applications of this skill.

Remember, the key to mastering this concept is practice. Work through plenty of examples, and don't be afraid to ask for help when you need it. With a little effort, you'll be simplifying radical expressions like a pro in no time! Keep practicing, and you'll find that these kinds of problems become much easier with time and experience. Good luck, and happy simplifying!