Math Problem Statement

Simplify the expression \( \frac{3}{2 + \sqrt{2}} \times \frac{2 - \sqrt{2}}{2 - \sqrt{2}} \).

Solution

In this image, you have the following mathematical expression:

32+2×2222\frac{3}{2 + \sqrt{2}} \times \frac{2 - \sqrt{2}}{2 - \sqrt{2}}

This expression involves simplifying the fraction by multiplying the numerator and denominator by the conjugate of the denominator. Let me help you simplify this step by step.

Step 1: Recognize the Conjugate

You are multiplying by the conjugate of 2+22 + \sqrt{2}, which is 222 - \sqrt{2}. This will help eliminate the square root from the denominator.

Step 2: Multiply Denominator by Conjugate

In the denominator, multiply: (2+2)(22)=22(2)2=42=2(2 + \sqrt{2})(2 - \sqrt{2}) = 2^2 - (\sqrt{2})^2 = 4 - 2 = 2 So the denominator becomes 2.

Step 3: Multiply Numerator

In the numerator, you have: 3×(22)=6323 \times (2 - \sqrt{2}) = 6 - 3\sqrt{2}

Step 4: Simplified Expression

Now, putting everything together, the expression simplifies to: 6322\frac{6 - 3\sqrt{2}}{2}

You can further simplify this by dividing both terms in the numerator by 2: 33223 - \frac{3\sqrt{2}}{2}

So the final simplified expression is: 33223 - \frac{3\sqrt{2}}{2}

Would you like more details on any step or have further questions?

Here are some related questions you might consider:

  1. What is the general form for simplifying expressions with square roots in the denominator?
  2. Why is it useful to multiply by the conjugate when simplifying radicals?
  3. Can this method be applied to more complex denominators involving other roots?
  4. How does the concept of rationalizing the denominator work in different algebraic contexts?
  5. How can the process of rationalizing be extended to solve equations with square roots?

Tip: When simplifying fractions involving square roots, always check if multiplying by the conjugate will help simplify the expression more efficiently.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rationalization
Radicals

Formulas

Conjugate multiplication: (a + b)(a - b) = a^2 - b^2
Rationalizing the denominator: Multiply by the conjugate of the denominator

Theorems

Conjugate theorem: The product of a number and its conjugate results in a rational number

Suitable Grade Level

Grades 9-12