Rationalize The Denominator And Simplify 9/sqrt(3)

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In the realm of mathematics, simplifying expressions is a fundamental skill. One common simplification technique is rationalizing the denominator. This process eliminates radicals (like square roots) from the denominator of a fraction, making it easier to work with and compare to other expressions. In this detailed guide, we will dive deep into the process of rationalizing the denominator, with a specific focus on simplifying the expression $\frac{9}{\sqrt{3}}$.

The core idea behind rationalizing the denominator is to manipulate a fraction so that the denominator is a rational number (i.e., a number that can be expressed as a fraction of two integers). This is often achieved by multiplying both the numerator and the denominator of the fraction by a suitable expression that eliminates the radical in the denominator. The expression we choose to multiply by is typically the radical itself or its conjugate (in the case of binomial denominators containing radicals).

Why do we rationalize the denominator? There are several reasons:

• Simplification: It makes the expression simpler to understand and work with. A rational denominator allows for easier comparison and combination of fractions.
• Standard Form: Mathematical convention generally favors expressions with rational denominators. It's considered a more refined and standard way of presenting a fraction.
• Calculations: Performing calculations, especially by hand, is often easier with a rational denominator.

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Let's apply the concept of rationalizing the denominator to the given expression, $\frac{9}{\sqrt{3}}$. Here's a step-by-step breakdown:

Step 1: Identify the Radical in the Denominator

The first step is to identify the radical in the denominator. In this case, it's $\sqrt{3}$.

Step 2: Determine the Rationalizing Factor

The rationalizing factor is the expression that, when multiplied by the denominator, will eliminate the radical. For a simple square root like $\sqrt{3}$, the rationalizing factor is $\sqrt{3}$ itself. Multiplying $\sqrt{3}$ by $\sqrt{3}$ gives us 3, which is a rational number.

Step 3: Multiply the Numerator and Denominator by the Rationalizing Factor

Multiply both the numerator and the denominator of the fraction by the rationalizing factor, $\sqrt{3}$:

$$\frac{9}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$$

Step 4: Perform the Multiplication

Multiply the numerators: $9 \times \sqrt{3} = 9\sqrt{3}$

Multiply the denominators: $\sqrt{3} \times \sqrt{3} = 3$

This gives us:

$$\frac{9\sqrt{3}}{3}$$

Step 5: Simplify the Result

Now, simplify the resulting fraction. Notice that both the numerator and the denominator have a common factor of 3. Divide both by 3:

$$\frac{9\sqrt{3}}{3} = \frac{3 \times 3\sqrt{3}}{3} = 3\sqrt{3}$$

Therefore, the simplified form of $\frac{9}{\sqrt{3}}$ after rationalizing the denominator is $3\sqrt{3}$.

Let's solidify our understanding with a few more examples of rationalizing the denominator:

Example 1: Simplify $\frac{4}{\sqrt{2}}$

1. Identify the radical: $\sqrt{2}$
2. Rationalizing factor: $\sqrt{2}$
3. Multiply: $\frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2}$
4. Simplify: $\frac{4\sqrt{2}}{2} = 2\sqrt{2}$

Thus, the simplified form is $2\sqrt{2}$.

Example 2: Simplify $\frac{1}{\sqrt{5}}$

1. Identify the radical: $\sqrt{5}$
2. Rationalizing factor: $\sqrt{5}$
3. Multiply: $\frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5}$
4. Simplify: The fraction is already in its simplest form.

Therefore, the simplified form is $\frac{\sqrt{5}}{5}$.

The process becomes slightly more involved when the denominator is a binomial expression containing radicals, such as $a + \sqrt{b}$ or $\sqrt{a} - \sqrt{b}$. In these cases, we use the conjugate of the denominator as the rationalizing factor.

The conjugate of a binomial expression $a + b$ is $a - b$, and vice versa. The key property of conjugates is that when multiplied, they eliminate the radical term due to the difference of squares identity: $(a + b)(a - b) = a^2 - b^2$.

Example 3: Simplify $\frac{1}{1 + \sqrt{2}}$

1. Identify the denominator: $1 + \sqrt{2}$
2. Find the conjugate: The conjugate of $1 + \sqrt{2}$ is $1 - \sqrt{2}$.
3. Multiply: $\frac{1}{1 + \sqrt{2}} \times \frac{1 - \sqrt{2}}{1 - \sqrt{2}}$
4. Multiply numerators: $1 \times (1 - \sqrt{2}) = 1 - \sqrt{2}$
5. Multiply denominators: $(1 + \sqrt{2})(1 - \sqrt{2}) = 1^2 - (\sqrt{2})^2 = 1 - 2 = -1$

This gives us:

$$\frac{1 - \sqrt{2}}{-1}$$

6. Simplify: Divide both numerator and denominator by -1:

$$\frac{1 - \sqrt{2}}{-1} = -1 + \sqrt{2} = \sqrt{2} - 1$$

Thus, the simplified form is $\sqrt{2} - 1$.

Example 4: Simplify $\frac{\sqrt{3}}{2 - \sqrt{3}}$

1. Identify the denominator: $2 - \sqrt{3}$
2. Find the conjugate: The conjugate of $2 - \sqrt{3}$ is $2 + \sqrt{3}$.
3. Multiply: $\frac{\sqrt{3}}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}}$
4. Multiply numerators: $\sqrt{3} \times (2 + \sqrt{3}) = 2\sqrt{3} + 3$
5. Multiply denominators: $(2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1$

This gives us:

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

6. Simplify: The fraction simplifies to $2\sqrt{3} + 3$.

Therefore, the simplified form is $2\sqrt{3} + 3$.

• Forgetting to multiply both the numerator and denominator: Always multiply both the top and bottom of the fraction by the rationalizing factor to maintain the fraction's value.
• Incorrectly identifying the conjugate: Ensure you correctly determine the conjugate, especially when dealing with binomial expressions.
• Not simplifying the result: After rationalizing, always check if the resulting fraction can be further simplified by canceling common factors.
• Applying the technique unnecessarily: Rationalizing the denominator is only necessary when there is a radical in the denominator. Don't apply it to fractions that already have rational denominators.

Rationalizing the denominator is a critical skill in algebra and beyond. It simplifies expressions, adheres to mathematical conventions, and makes calculations more manageable. By understanding the underlying principles and practicing with various examples, you can confidently rationalize any denominator and simplify complex expressions. In the specific case of $\frac{9}{\sqrt{3}}$, we demonstrated the step-by-step process, ultimately arriving at the simplified form of $3\sqrt{3}$. Keep practicing, and you'll master this essential technique in no time!

By grasping this concept, you'll not only excel in your current studies but also build a strong foundation for more advanced mathematical concepts in the future. Remember, mathematics is a journey, and every step you take strengthens your understanding and problem-solving abilities.