Question - Simplifying a Fraction Using Difference of Squares
Solution:
The expression in the image appears to be a fraction that needs to be simplified:(1 - √2)/(√5 - √3) - (1 + √2)/(√5 + √3)To simplify this, we'll use the difference of squares technique. Here's the step-by-step process:1. Multiply the numerator and denominator of each fraction by the conjugate of the respective denominator.For the first fraction, the conjugate of (√5 - √3) is (√5 + √3), and for the second fraction, the conjugate of (√5 + √3) is (√5 - √3).So, we have:(1 - √2)(√5 + √3) / [(√5 - √3)(√5 + √3)] - (1 + √2)(√5 - √3) / [(√5 + √3)(√5 - √3)]2. Apply the difference of squares to the denominators:(√5)^2 - (√3)^2 = 5 - 3 = 2.3. Expand the numerators:(1 - √2)(√5 + √3) = √5 + √3 - √10 - √6(1 + √2)(√5 - √3) = √5 - √3 + √10 - √64. Now we have two fractions with the same denominator, which allows us to combine them:(√5 + √3 - √10 - √6) - (√5 - √3 + √10 - √6) / 25. Combine like terms in the numerator:The √5 and -√5 cancel each other out, the √10 and -√10 cancel each other out, and we're left with:(√3 - √3 - √6 + √6) / 26. Simplify the remaining terms:All terms in the numerator cancel out, leaving us with 0/2.7. Therefore, the simplified form of the given expression is:0The entire simplified result is 0.
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