Simplify The Expression: 3 3 − 3 5 \frac{3}{3 - 3 \sqrt{5}} 3 − 3 5 ​ 3 ​

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Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially when dealing with expressions involving square roots. In this article, we will focus on simplifying the given expression $\frac{3}{3 - 3 \sqrt{5}}$ by rationalizing the denominator. This process will help us eliminate the radical from the denominator, making it easier to work with the expression.

Understanding the Concept of Rationalizing the Denominator

Rationalizing the denominator is a technique used to remove radicals from the denominator of a fraction. This is achieved by multiplying both the numerator and the denominator by a suitable expression that eliminates the radical. The goal is to simplify the expression and make it easier to work with.

Step 1: Identify the Radical in the Denominator

The given expression $\frac{3}{3 - 3 \sqrt{5}}$ has a radical in the denominator, which is $\sqrt{5}$. To rationalize the denominator, we need to eliminate this radical.

Step 2: Determine the Suitable Expression to Multiply

To eliminate the radical $\sqrt{5}$, we need to multiply both the numerator and the denominator by an expression that will result in the radical being removed. In this case, we can multiply by the conjugate of the denominator, which is $3 + 3 \sqrt{5}$.

Step 3: Multiply the Numerator and Denominator by the Conjugate

We multiply both the numerator and the denominator by $3 + 3 \sqrt{5}$:

$$\frac{3}{3 - 3 \sqrt{5}} \cdot \frac{3 + 3 \sqrt{5}}{3 + 3 \sqrt{5}}$$

Step 4: Simplify the Expression

Now, we simplify the expression by multiplying the numerators and denominators:

$$\frac{3(3 + 3 \sqrt{5})}{(3 - 3 \sqrt{5})(3 + 3 \sqrt{5})}$$

Step 5: Apply the Difference of Squares Formula

We can simplify the denominator using the difference of squares formula:

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

In this case, we have:

$$(3 - 3 \sqrt{5})(3 + 3 \sqrt{5}) = 3^2 - (3 \sqrt{5})^2$$

Step 6: Simplify the Denominator

Now, we simplify the denominator:

$$3^2 - (3 \sqrt{5})^2 = 9 - 45 = -36$$

Step 7: Simplify the Numerator

We also simplify the numerator:

$$3(3 + 3 \sqrt{5}) = 9 + 9 \sqrt{5}$$

Step 8: Write the Final Simplified Expression

Now, we can write the final simplified expression:

$$\frac{9 + 9 \sqrt{5}}{-36}$$

Conclusion

In this article, we simplified the expression $\frac{3}{3 - 3 \sqrt{5}}$ by rationalizing the denominator. We identified the radical in the denominator, determined the suitable expression to multiply, and multiplied the numerator and denominator by the conjugate. We then simplified the expression using the difference of squares formula and wrote the final simplified expression.

Final Answer

The final simplified expression is:

$$\frac{9 + 9 \sqrt{5}}{-36}$$

Related Topics

• Rationalizing the denominator
• Simplifying complex fractions
• Difference of squares formula

Example Problems

• Simplify the expression $\frac{2}{2 - 2 \sqrt{3}}$
• Rationalize the denominator of the expression $\frac{4}{4 + 4 \sqrt{2}}$

Practice Problems

• Simplify the expression $\frac{1}{1 - 1 \sqrt{2}}$
• Rationalize the denominator of the expression $\frac{3}{3 - 3 \sqrt{3}}$

References

• [1] "Rationalizing the Denominator" by Math Open Reference
• [2] "Simplifying Complex Fractions" by Khan Academy
  

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Introduction

In our previous article, we simplified the expression $\frac{3}{3 - 3 \sqrt{5}}$ by rationalizing the denominator. In this article, we will answer some frequently asked questions related to rationalizing the denominator and simplifying complex fractions.

Q&A

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is a technique used to remove radicals from the denominator of a fraction. This is achieved by multiplying both the numerator and the denominator by a suitable expression that eliminates the radical.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to simplify the expression and make it easier to work with. Rationalizing the denominator helps to eliminate the radical from the denominator, making it easier to perform operations such as addition, subtraction, multiplication, and division.

Q: How do we rationalize the denominator?

A: To rationalize the denominator, we need to multiply both the numerator and the denominator by an expression that will result in the radical being removed. In most cases, we can multiply by the conjugate of the denominator.

Q: What is the conjugate of the denominator?

A: The conjugate of the denominator is an expression that, when multiplied by the denominator, will result in the radical being removed. For example, if the denominator is $a - b$, the conjugate is $a + b$.

Q: Can we rationalize the denominator of a fraction with a negative exponent?

A: Yes, we can rationalize the denominator of a fraction with a negative exponent. However, we need to be careful when multiplying the numerator and denominator by the conjugate, as this may result in a negative exponent.

Q: How do we simplify complex fractions?

A: To simplify complex fractions, we need to multiply both the numerator and the denominator by an expression that will result in the radical being removed. We can then simplify the expression using the difference of squares formula.

Q: What is the difference of squares formula?

A: The difference of squares formula is a mathematical formula that states:

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

This formula can be used to simplify expressions involving the product of two binomials.

Q: Can we simplify complex fractions with multiple radicals?

A: Yes, we can simplify complex fractions with multiple radicals. However, we need to be careful when multiplying the numerator and denominator by the conjugate, as this may result in multiple radicals.

Q: How do we determine the suitable expression to multiply?

A: To determine the suitable expression to multiply, we need to identify the radical in the denominator and determine the conjugate of the denominator. We can then multiply both the numerator and the denominator by the conjugate.

Q: Can we rationalize the denominator of a fraction with a variable in the denominator?

A: Yes, we can rationalize the denominator of a fraction with a variable in the denominator. However, we need to be careful when multiplying the numerator and denominator by the conjugate, as this may result in a variable in the denominator.

Q: How do we simplify expressions involving the product of two binomials?

A: To simplify expressions involving the product of two binomials, we can use the difference of squares formula. This formula states:

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

We can then simplify the expression using this formula.

Conclusion

In this article, we answered some frequently asked questions related to rationalizing the denominator and simplifying complex fractions. We discussed the concept of rationalizing the denominator, the difference of squares formula, and how to simplify complex fractions with multiple radicals.

Final Answer

The final answer to the question of how to simplify the expression $\frac{3}{3 - 3 \sqrt{5}}$ is:

$$\frac{9 + 9 \sqrt{5}}{-36}$$

Related Topics

• Rationalizing the denominator
• Simplifying complex fractions
• Difference of squares formula

Example Problems

• Simplify the expression $\frac{2}{2 - 2 \sqrt{3}}$
• Rationalize the denominator of the expression $\frac{4}{4 + 4 \sqrt{2}}$

Practice Problems

• Simplify the expression $\frac{1}{1 - 1 \sqrt{2}}$
• Rationalize the denominator of the expression $\frac{3}{3 - 3 \sqrt{3}}$

References

• [1] "Rationalizing the Denominator" by Math Open Reference
• [2] "Simplifying Complex Fractions" by Khan Academy