Examkin

Question

Difficulty: MediumSurds and Rationalisation

If 35−35+3=a+b15\frac{3\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} = a + b\sqrt{15}, where aa and bb are rational numbers, what is the value of a−ba - b?

  1. 11Answer
  2. B
    8
  3. C
    5
  4. D
    13

Answer

11
By multiplying the numerator and denominator by the conjugate of the denominator, (5−3)(\sqrt{5} - \sqrt{3}), the fraction simplifies to 18−4152=9−215\frac{18 - 4\sqrt{15}}{2} = 9 - 2\sqrt{15}. Equating this to a+b15a + b\sqrt{15} gives a=9a = 9 and b=−2b = -2. Calculating a−ba - b gives 9−(−2)=119 - (-2) = 11.

Step-by-Step Solution

1
Multiply the numerator and denominator by the conjugate of the denominator
\frac{3\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \times \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}} = \frac{(3\sqrt{5} - \sqrt{3})(\sqrt{5} - \sqrt{3})}{(\sqrt{5})^2 - (\sqrt{3})^2}
Rationalising the denominator requires using the difference of squares identity (x+y)(x−y)=x2−y2(x+y)(x-y) = x^2 - y^2.
2
Expand the numerator and denominator
Denominator = 5 - 3 = 2. Numerator = 3(5) - 3\sqrt{15} - \sqrt{15} + 3 = 15 + 3 - 4\sqrt{15} = 18 - 4\sqrt{15}.
Apply distributive property to expand (35−3)(5−3)(3\sqrt{5} - \sqrt{3})(\sqrt{5} - \sqrt{3}) carefully combining like terms.
3
Simplify the fraction to match the form a+b15a + b\sqrt{15}
\frac{18 - 4\sqrt{15}}{2} = 9 - 2\sqrt{15}
Divide each term in the numerator by 2.
4
Identify aa and bb and evaluate a−ba - b
a = 9, b = -2 \implies a - b = 9 - (-2) = 11
Subtracting negative 2 is equivalent to adding 2.

Key Concept

Rationalisation of Binomial Surd Denominators
Estimated Time:1m 30s
Rate this question

Topics

Surds and Rationalisation

More questions from this topic