We are given the expression:

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

To simplify this expression, we will rationalize the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator, which is $ 2 + \sqrt{3} $.

### Step 1: Multiply by the conjugate
We multiply both the numerator and denominator by $ 2 + \sqrt{3} $:

\[
y = \frac{\sqrt{2 + \sqrt{3}}}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}}
\]

This gives:

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

### Step 2: Simplify the denominator
We now simplify the denominator using the difference of squares:

\[
(2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1
\]

So the denominator is 1, and we are left with:

\[
y = \sqrt{2 + \sqrt{3}}(2 + \sqrt{3})
\]

### Step 3: Expand the expression
Now expand the numerator:

\[
y = \sqrt{2 + \sqrt{3}} \times (2 + \sqrt{3})
\]

This is as simplified as it gets without knowing more about the numerical value of $ \sqrt{2 + \sqrt{3}} $, so the final expression is:

\[
y = \sqrt{2 + \sqrt{3}}(2 + \sqrt{3})
\]

Let me know if you'd like further details or explanations!

### Related Questions:
1. How do you find the conjugate of a binomial with radicals?
2. What are the steps to rationalize a denominator containing a square root?
3. How do you simplify an expression with nested square roots?
4. What is the difference of squares and how is it used in rationalizing denominators?
5. Can you apply these steps to other expressions with radicals?

### Tip:
Always check if multiplying by the conjugate simplifies the denominator completely or requires additional steps!

Learn how to simplify the expression $$ y=\frac{\sqrt{2+\sqrt3}}{2-\sqrt3} $$ by rationalizing the denominator and applying the difference of squares. This detailed solution breaks down key algebraic concepts involving radicals and rationalization. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation