https://file.aiquickdraw.com/mathbot/file/1726533312255afk5m76m.jpg

The problem asks to simplify the expression \( \frac{5}{2 + \sqrt{3}} \) by rationalizing the denominator. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator, \( 2 - \sqrt{3} \). Let's work through the steps:

\[
\frac{5}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{5(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})}
\]

### Step-by-step:

1. **Multiply the numerator**:
   \[
   5(2 - \sqrt{3}) = 10 - 5\sqrt{3}
   \]

2. **Simplify the denominator**:
   The denominator is a difference of squares:
   \[
   (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1
   \]

3. **Final simplified form**:
   \[
   \frac{10 - 5\sqrt{3}}{1} = 10 - 5\sqrt{3}
   \]

Therefore, the simplified expression is \( 10 - 5\sqrt{3} \), which corresponds to option **e**.

Let me know if you'd like a more detailed explanation!

---

Here are five related questions to expand your understanding:

1. How do you rationalize a denominator when it contains a square root?
2. What is the difference between multiplying by the conjugate and regular multiplication?
3. How does the difference of squares formula help in simplifying irrational expressions?
4. Can you give an example of rationalizing a denominator with more complex roots?
5. Why is rationalizing the denominator useful in algebraic expressions?

**Tip:** Always remember that the conjugate of \( a + b\sqrt{n} \) is \( a - b\sqrt{n} \), which is key to eliminating square roots from the denominator.

Dengan merasionalkan penyebut, bentuk 5/(2 + √3) dapat disederhanakan menjadi.....

Learn how to simplify the expression 5/(2 + √3) by rationalizing the denominator using the difference of squares and multiplying by the conjugate. This problem covers key algebraic techniques, including simplifying radical expressions. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation