Math Problem Statement

√(4 - √3) · √(4 + √3) işleminin sonucu kaçtır?

Solution

To solve the given expression, let's break it down step by step.

Problem:

434+3\sqrt{4 - \sqrt{3}} \cdot \sqrt{4 + \sqrt{3}}

Solution:

  1. Use the property of square roots: ab=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}

    Applying this property: 434+3=(43)(4+3)\sqrt{4 - \sqrt{3}} \cdot \sqrt{4 + \sqrt{3}} = \sqrt{(4 - \sqrt{3}) \cdot (4 + \sqrt{3})}

  2. Expand (43)(4+3)(4 - \sqrt{3})(4 + \sqrt{3}) using the difference of squares formula: (ab)(a+b)=a2b2(a - b)(a + b) = a^2 - b^2 Here, a=4a = 4 and b=3b = \sqrt{3}. Substituting these values: (43)(4+3)=42(3)2(4 - \sqrt{3})(4 + \sqrt{3}) = 4^2 - (\sqrt{3})^2

  3. Simplify: 42=16and(3)2=34^2 = 16 \quad \text{and} \quad (\sqrt{3})^2 = 3 So: 163=1316 - 3 = 13

  4. Return to the square root: (43)(4+3)=13\sqrt{(4 - \sqrt{3})(4 + \sqrt{3})} = \sqrt{13}

Final Answer:

The result of the expression is: 13\boxed{\sqrt{13}}


Would you like further clarification or additional examples?
Here are 5 related questions for further exploration:

  1. How does the difference of squares formula simplify similar expressions?
  2. Can the square root of 13 be approximated to a decimal value?
  3. What happens if the product includes complex numbers?
  4. How would this problem change if a different term replaced 3\sqrt{3}?
  5. What are other ways to simplify products of square roots?

Tip: Always look for the possibility of applying algebraic identities like the difference of squares when dealing with nested square root expressions!

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Math Problem Analysis

Mathematical Concepts

Square roots
Difference of squares

Formulas

√a · √b = √(a · b)
(a - b)(a + b) = a^2 - b^2

Theorems

Difference of squares formula

Suitable Grade Level

Grades 7-9