Math Problem Statement
Solution
To solve the given problem, we aim to rewrite:
in the form , where and are integers. Let's break it down step by step.
Step 1: Simplify the numerator
The numerator is . Notice that:
So, the numerator becomes:
Step 2: Rationalize the denominator
The denominator is . To rationalize it, multiply both the numerator and denominator by the conjugate of the denominator, :
Step 3: Simplify the denominator
Using the difference of squares formula:
So, the denominator simplifies to .
Step 4: Expand the numerator
Now expand using the distributive property:
Simplify each term:
Since , this becomes:
Combine like terms:
Step 5: Divide by the denominator
Now divide the expanded numerator by the denominator, :
Distribute the division:
Final Answer:
The expression simplifies to:
where and .
Would you like me to explain any step further or assist with a related problem? Here are some related questions to consider:
- What is the general process for rationalizing a denominator with a square root?
- How do you simplify square roots such as ?
- Why does multiplying by the conjugate work to rationalize denominators?
- Can all expressions of this form be rewritten as ?
- How would the result change if the denominator was ?
Tip: Always simplify square roots at the earliest step to make calculations easier!
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Math Problem Analysis
Mathematical Concepts
Simplification of algebraic expressions
Rationalization of denominators
Simplifying square roots
Formulas
Difference of squares: (a - b)(a + b) = a^2 - b^2
Square root simplification: √(mn) = √m√n
Theorems
Rationalization of denominators using conjugates
Suitable Grade Level
Grades 9-12
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