https://r2.erweima.ai/mathbot/image/17235595630646h4tqkgc.jpg

Let's solve the expression given in the problem:

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

### Step 1: Simplify the square root expression
The term inside the square root is:

\[
\sqrt{\frac{2 + \sqrt{3}}{2 - \sqrt{3}}}
\]

We can simplify this by multiplying the numerator and denominator by the conjugate of the denominator:

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

Simplifying further:

\[
(2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3}
\]

And the denominator:

\[
4 - 3 = 1
\]

So the expression simplifies to:

\[
\sqrt{7 + 4\sqrt{3}}
\]

### Step 2: Substitute back into the original expression
Now the expression becomes:

\[
-(2 - \sqrt{3}) \cdot \sqrt{7 + 4\sqrt{3}} + 5
\]

### Step 3: Simplify further
We observe that:

\[
\sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3}
\]

So, substituting this in:

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

Expand the multiplication:

\[
-(4 - 3) + 5 = -(1) + 5 = 4
\]

### Final Answer:
The solution to the problem is **4**, which corresponds to option **e)**.

Would you like further details or have any questions?

---

Here are 8 related questions:
1. How do you find the conjugate of a binomial expression?
2. What is the significance of rationalizing the denominator?
3. How can you simplify expressions involving square roots?
4. What are some common mistakes to avoid when simplifying radical expressions?
5. How do you multiply binomials involving radicals?
6. Can you explain why the conjugate is used in simplifying expressions like this?
7. How would you handle similar expressions with different coefficients?
8. What are the general steps for solving complex expressions like this one?

**Tip:** Always remember to multiply by the conjugate when dealing with expressions that contain a radical in the denominator—this helps to simplify the expression.

Learn how to simplify and solve a complex square root expression involving radicals. Step-by-step explanation provided, including the use of conjugates and simplification techniques. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation