Simplifying The Quotient: (9+√2) / (4-√7) Explained

Have you ever stumbled upon a mathematical expression that looks a bit intimidating? Expressions involving quotients with radicals, like (9+√2) / (4-√7), can seem tricky at first glance. But don't worry! This comprehensive guide breaks down the steps to simplify this type of expression, making it easy to understand and solve. Let’s dive in and unravel the mystery behind simplifying this quotient.

Understanding the Problem

Before we jump into the solution, let's first understand what we're dealing with. The expression $\frac{9+\sqrt{2}}{4-\sqrt{7}}$ is a quotient, which means it's a division problem. The numerator (the top part) is $9+\sqrt{2}$, and the denominator (the bottom part) is $4-\sqrt{7}$. The presence of square roots (radicals) in both the numerator and the denominator is what makes this problem a bit more complex.

The key to simplifying this type of expression lies in a technique called rationalizing the denominator. Rationalizing the denominator means getting rid of the radical in the denominator. We achieve this by multiplying both the numerator and the denominator by the conjugate of the denominator. But what is a conjugate? Let's find out.

What is a Conjugate?

In mathematics, the conjugate of a binomial expression (an expression with two terms) that includes a radical is formed by changing the sign between the terms. For example, if we have an expression like $a + \sqrt{b}$, its conjugate is $a - \sqrt{b}$. Similarly, the conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$.

So, what is the conjugate of our denominator, $4-\sqrt{7}$? It's simply $4+\sqrt{7}$. We'll use this conjugate to rationalize the denominator in the next step. Understanding conjugates is crucial in simplifying expressions like these, as it allows us to eliminate the radical from the denominator, making the expression easier to work with and understand. By multiplying the denominator by its conjugate, we utilize the difference of squares pattern, which results in a rational number.

The Process of Rationalizing the Denominator

Now that we know about conjugates, let's get back to our original problem: $\frac{9+\sqrt{2}}{4-\sqrt{7}}$. To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator, which is $4+\sqrt{7}$.

Here's how it looks:

$$\frac{9+\sqrt{2}}{4-\sqrt{7}} \times \frac{4+\sqrt{7}}{4+\sqrt{7}}$$

Multiplying the Numerators

First, let's multiply the numerators: $(9+\sqrt{2}) \times (4+\sqrt{7})$. We'll use the distributive property (also known as the FOIL method) to expand this:

• $9 \times 4 = 36$
• $9 \times \sqrt{7} = 9\sqrt{7}$
• $\sqrt{2} \times 4 = 4\sqrt{2}$
• $\sqrt{2} \times \sqrt{7} = \sqrt{14}$

So, the numerator becomes: $36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}$.

Multiplying the Denominators

Next, let's multiply the denominators: $(4-\sqrt{7}) \times (4+\sqrt{7})$. This is where the magic of conjugates comes into play. When we multiply a binomial by its conjugate, we get the difference of squares:

$(a - b)(a + b) = a^2 - b^2$

In our case, $a = 4$ and $b = \sqrt{7}$. So, the denominator becomes:

$4^2 - (\sqrt{7})^2 = 16 - 7 = 9$

Notice that the radical has disappeared from the denominator! This is exactly what we wanted to achieve.

Putting It All Together

Now, let's put the simplified numerator and denominator back together:

$$\frac{36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}}{9}$$

This is the simplified form of the quotient. While it might still look a bit complex, we've successfully eliminated the radical from the denominator, which is a significant step in simplifying the expression. Remember, the process of rationalizing the denominator involves multiplying both the numerator and the denominator by the conjugate of the denominator. This technique is widely used in mathematics to simplify expressions and make them easier to work with.

Final Simplified Expression

So, after rationalizing the denominator, the simplified form of the quotient $\frac{9+\sqrt{2}}{4-\sqrt{7}}$ is:

$$\frac{36 + 9\sqrt{7} + 4\sqrt{2} + \sqrt{14}}{9}$$

We can also express this as:

$$4 + \sqrt{7} + \frac{4\sqrt{2}}{9} + \frac{\sqrt{14}}{9}$$

This final expression is now in a simplified form, with no radicals in the denominator. This makes it easier to understand and use in further calculations.

Why Rationalize the Denominator?

You might be wondering, why do we even bother rationalizing the denominator? There are several reasons why this technique is important:

1. Simplification: It simplifies the expression, making it easier to work with. An expression with a rational denominator is generally considered to be in a simpler form.
2. Comparison: It makes it easier to compare different expressions. If you have two expressions with radicals in the denominators, it's difficult to compare them directly. Rationalizing the denominators makes comparison straightforward.
3. Further Calculations: It simplifies further calculations. Expressions with rational denominators are easier to manipulate in subsequent algebraic steps.
4. Convention: It's a mathematical convention. In mathematics, it's generally preferred to have expressions with rational denominators.

The process of rationalizing the denominator is a fundamental technique in algebra and is widely used in various mathematical contexts. It not only simplifies expressions but also makes them more manageable for further analysis and calculations.

Common Mistakes to Avoid

When simplifying quotients with radicals, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions.

1. Forgetting to Multiply Both Numerator and Denominator: One of the most common mistakes is multiplying only the denominator by the conjugate. Remember, to maintain the value of the expression, you must multiply both the numerator and the denominator by the same value.
2. Incorrectly Multiplying the Numerators: When multiplying the numerators, it's crucial to use the distributive property (FOIL method) correctly. Make sure to multiply each term in the first binomial by each term in the second binomial.
3. Mistaking the Conjugate: The conjugate is formed by changing the sign between the terms in the denominator. For example, the conjugate of $a - \sqrt{b}$ is $a + \sqrt{b}$. Ensure you identify the correct conjugate before multiplying.
4. Incorrectly Simplifying Radicals: Sometimes, after multiplying, you might need to simplify the radicals further. Make sure to look for perfect square factors within the radicals and simplify them accordingly.
5. Not Recognizing the Difference of Squares: When multiplying the denominator by its conjugate, remember to use the difference of squares pattern: $(a - b)(a + b) = a^2 - b^2$. This will help you quickly eliminate the radical from the denominator.
6. Skipping Steps: It's tempting to skip steps to save time, but this can lead to errors. Take your time and write out each step clearly, especially when you're first learning this technique.

By being mindful of these common mistakes, you can improve your accuracy and confidence in simplifying quotients with radicals. Practice makes perfect, so the more you work through these types of problems, the better you'll become.

Practice Problems

To solidify your understanding of simplifying quotients with radicals, let's work through a couple of practice problems.

Problem 1: Simplify the expression $\frac{5}{3 + \sqrt{2}}$

Solution:

1. Identify the conjugate of the denominator: The conjugate of $3 + \sqrt{2}$ is $3 - \sqrt{2}$.
2. Multiply both the numerator and the denominator by the conjugate:

   $$\frac{5}{3 + \sqrt{2}} \times \frac{3 - \sqrt{2}}{3 - \sqrt{2}}$$

3. Multiply the numerators: $5 \times (3 - \sqrt{2}) = 15 - 5\sqrt{2}$
4. Multiply the denominators: $(3 + \sqrt{2})(3 - \sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7$
5. Write the simplified expression:

   $$\frac{15 - 5\sqrt{2}}{7}$$

Problem 2: Simplify the expression $\frac{1 + \sqrt{3}}{2 - \sqrt{3}}$

Solution:

1. Identify the conjugate of the denominator: The conjugate of $2 - \sqrt{3}$ is $2 + \sqrt{3}$.
2. Multiply both the numerator and the denominator by the conjugate:

   $$\frac{1 + \sqrt{3}}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}}$$

3. Multiply the numerators: $(1 + \sqrt{3})(2 + \sqrt{3}) = 1 \times 2 + 1 \times \sqrt{3} + \sqrt{3} \times 2 + \sqrt{3} \times \sqrt{3} = 2 + \sqrt{3} + 2\sqrt{3} + 3 = 5 + 3\sqrt{3}$
4. Multiply the denominators: $(2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1$
5. Write the simplified expression:

   $$\frac{5 + 3\sqrt{3}}{1} = 5 + 3\sqrt{3}$$

By working through these practice problems, you can see how the process of rationalizing the denominator works in action. Remember to identify the conjugate, multiply both the numerator and the denominator, and simplify the resulting expression.

Conclusion

Simplifying quotients with radicals might seem challenging initially, but with a clear understanding of the steps involved, it becomes quite manageable. The key is to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. This technique eliminates the radical from the denominator, resulting in a simplified expression. Remember to use the distributive property (FOIL method) when multiplying binomials and the difference of squares pattern when multiplying conjugates.

By following the steps outlined in this guide and practicing with various examples, you'll become proficient in simplifying quotients with radicals. This skill is essential in algebra and will be helpful in more advanced mathematical concepts. So, keep practicing, and don't hesitate to revisit this guide whenever you need a refresher.

For further learning and exploration on this topic, you can visit resources like Khan Academy's Algebra section.