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Difficulty: MediumSurds and Rationalisation

Simplify the expression 33−1−33+1\frac{\sqrt{3}}{\sqrt{3} - 1} - \frac{\sqrt{3}}{\sqrt{3} + 1}.

  1. 3\sqrt{3}Answer
  2. B
    232\sqrt{3}
  3. C
    33
  4. D
    00

Answer

3\sqrt{3}
Combining the fractions using the common denominator (3−1)(3+1)=2(\sqrt{3}-1)(\sqrt{3}+1) = 2 gives a numerator of 3(3+1)−3(3−1)=3+3−3+3=23\sqrt{3}(\sqrt{3}+1) - \sqrt{3}(\sqrt{3}-1) = 3 + \sqrt{3} - 3 + \sqrt{3} = 2\sqrt{3}. Dividing 232\sqrt{3} by 22 yields the simplified answer 3\sqrt{3}.

Step-by-Step Solution

1
Find a common denominator for the two fractions
The common denominator is (3−1)(3+1)=(3)2−(1)2=3−1=2(\sqrt{3} - 1)(\sqrt{3} + 1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2
The denominators are conjugate surds, so their product simplifies to a rational number using the difference of two squares.
2
Express the numerator over the common denominator
Numerator =3(3+1)−3(3−1)= \sqrt{3}(\sqrt{3} + 1) - \sqrt{3}(\sqrt{3} - 1)
Multiply each numerator by the missing factor of the common denominator.
3
Expand and simplify the numerator
Numerator =(3+3)−(3−3)=3+3−3+3=23= (3 + \sqrt{3}) - (3 - \sqrt{3}) = 3 + \sqrt{3} - 3 + \sqrt{3} = 2\sqrt{3}
Distribute 3\sqrt{3} and handle the subtraction sign carefully.
4
Divide the simplified numerator by the common denominator
232=3\frac{2\sqrt{3}}{2} = \sqrt{3}
Cancel the common factor of 2.

Key Concept

Rationalisation of denominators and algebraic manipulation of surd fractions
Estimated Time:1m 15s
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Topics

Surds and Rationalisation

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