Math Problem Statement
Solution
In this image, you have the following mathematical expression:
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 , which is . This will help eliminate the square root from the denominator.
Step 2: Multiply Denominator by Conjugate
In the denominator, multiply: So the denominator becomes 2.
Step 3: Multiply Numerator
In the numerator, you have:
Step 4: Simplified Expression
Now, putting everything together, the expression simplifies to:
You can further simplify this by dividing both terms in the numerator by 2:
So the final simplified expression is:
Would you like more details on any step or have further questions?
Here are some related questions you might consider:
- What is the general form for simplifying expressions with square roots in the denominator?
- Why is it useful to multiply by the conjugate when simplifying radicals?
- Can this method be applied to more complex denominators involving other roots?
- How does the concept of rationalizing the denominator work in different algebraic contexts?
- 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
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