Let's evaluate the expression \(\sqrt{7 + 4\sqrt{3}} - \sqrt{7 - 4\sqrt{3}}\).

### Step 1: Define the expression
Let:
\[
x = \sqrt{7 + 4\sqrt{3}} - \sqrt{7 - 4\sqrt{3}}
\]

### Step 2: Square both sides
Square both sides to eliminate the square roots:
\[
x^2 = \left(\sqrt{7 + 4\sqrt{3}} - \sqrt{7 - 4\sqrt{3}}\right)^2
\]

Use the identity \((a - b)^2 = a^2 - 2ab + b^2\):
\[
x^2 = \left(\sqrt{7 + 4\sqrt{3}}\right)^2 - 2\sqrt{(7 + 4\sqrt{3})(7 - 4\sqrt{3})} + \left(\sqrt{7 - 4\sqrt{3}}\right)^2
\]

### Step 3: Simplify each term
1. Simplify \(\left(\sqrt{7 + 4\sqrt{3}}\right)^2\) and \(\left(\sqrt{7 - 4\sqrt{3}}\right)^2\):
   \[
   \left(\sqrt{7 + 4\sqrt{3}}\right)^2 = 7 + 4\sqrt{3}
   \]
   \[
   \left(\sqrt{7 - 4\sqrt{3}}\right)^2 = 7 - 4\sqrt{3}
   \]

2. Simplify the middle term:
   \[
   2\sqrt{(7 + 4\sqrt{3})(7 - 4\sqrt{3})}
   \]
   Using the difference of squares formula, \((a + b)(a - b) = a^2 - b^2\):
   \[
   (7 + 4\sqrt{3})(7 - 4\sqrt{3}) = 7^2 - (4\sqrt{3})^2 = 49 - 48 = 1
   \]
   So the middle term becomes:
   \[
   2\sqrt{1} = 2
   \]

### Step 4: Combine and simplify
Substitute the simplified expressions back into the equation:
\[
x^2 = (7 + 4\sqrt{3}) + (7 - 4\sqrt{3}) - 2
\]
\[
x^2 = 7 + 7 + 4\sqrt{3} - 4\sqrt{3} - 2 = 14 - 2 = 12
\]
So:
\[
x^2 = 12
\]
\[
x = \sqrt{12} = 2\sqrt{3}
\]

### Final Answer:
\[
\sqrt{7 + 4\sqrt{3}} - \sqrt{7 - 4\sqrt{3}} = 2\sqrt{3}
\]

Would you like any more details or have any other questions?

### Related Questions:
1. How do you simplify expressions involving nested square roots?
2. What is the general method for solving \(\sqrt{a + b\sqrt{c}}\) type expressions?
3. How does the difference of squares formula help in simplifying radical expressions?
4. Can the method used here be applied to other similar expressions?
5. How can you determine if a square root expression is rational?

### Tip:
When dealing with square roots in expressions, always check if the expression can be simplified using algebraic identities or special factorizations.

Explore how to evaluate the expression sqrt(7 + 4sqrt(3)) - sqrt(7 - 4sqrt(3)) step-by-step. Learn the difference of squares formula and simplify square root expressions. Suitable for high school students and anyone interested in algebraic identities.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation