Math Problem Statement
Rationalize the denominator of $\displaystyle \frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{7}}$, and write your answer in the form[ \frac{A\sqrt{2} + B\sqrt{3} + C\sqrt{7} + D\sqrt{E}}{F}, ]where everything is in simplest radical form and the fraction is in lowest terms, and $F$ is positive. What is $A + B + C + D + E + F$?
Solution
We are tasked with rationalizing the denominator of the expression
and expressing the result in the form
Step 1: Multiply by the Conjugate
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of . The conjugate in this case is . This will help eliminate the square roots in the denominator when simplified.
Thus, we multiply the expression by
This gives
Step 2: Simplify the Denominator
Now, we simplify the denominator. This is a difference of squares:
First, calculate :
Now, subtract :
Thus, the denominator simplifies to .
Step 3: Simplify the Expression
We now have:
We can factor out a 2 from the denominator:
Now, multiply both the numerator and denominator by (the conjugate of ) to further simplify:
Step 4: Simplify the Denominator Again
The denominator becomes:
Thus, the denominator is .
Step 5: Expand the Numerator
We now expand the numerator :
Simplifying, this becomes:
Step 6: Final Expression
The overall expression becomes:
[ \frac{-\sqrt{2} - \sqrt{6}\sqrt{2} - \sqrt{3} - \sqrt{6}\sqrt{3} + \sqrt{7} + \sqrt
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Math Problem Analysis
Mathematical Concepts
Algebra
Radical Expressions
Conjugates
Simplification
Formulas
Difference of squares: (a + b)(a - b) = a^2 - b^2
Multiplication of binomials: (x + y)^2
Theorems
Difference of squares theorem
Suitable Grade Level
Grades 10-12
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