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Difficulty: HardSurds and Rationalization of Denominators

If 32+2332−23\frac{3\sqrt{2} + 2\sqrt{3}}{3\sqrt{2} - 2\sqrt{3}} is expressed in the simplified form a+b6a + b\sqrt{6}, where aa and bb are rational numbers, what is the value of a+ba + b?

  1. A
    5
  2. 7Answer
  3. C
    9
  4. D
    13

Answer

The correct value of a+ba + b is 7.
Multiplying by the conjugate (32+23)(3\sqrt{2} + 2\sqrt{3}) reduces the denominator to 18−12=618 - 12 = 6 and expands the numerator to 30+12630 + 12\sqrt{6}. Dividing by 6 yields 5+265 + 2\sqrt{6}, giving a=5a = 5 and b=2b = 2, which sums to 7.

Step-by-Step Solution

1
Multiply the numerator and the denominator by the conjugate of the denominator, (32+23)(3\sqrt{2} + 2\sqrt{3}).
\frac{(3\sqrt{2} + 2\sqrt{3})(3\sqrt{2} + 2\sqrt{3})}{(3\sqrt{2} - 2\sqrt{3})(3\sqrt{2} + 2\sqrt{3})}
Rationalizing eliminates the surd terms from the denominator using the difference of two squares.
2
Expand both numerator and denominator.
Denominator: (32)2−(23)2=18−12=6(3\sqrt{2})^2 - (2\sqrt{3})^2 = 18 - 12 = 6. Numerator: (32)2+2(32)(23)+(23)2=18+126+12=30+126(3\sqrt{2})^2 + 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 18 + 12\sqrt{6} + 12 = 30 + 12\sqrt{6}.
Apply algebraic identities (x−y)(x+y)=x2−y2(x-y)(x+y) = x^2 - y^2 and (x+y)2=x2+2xy+y2(x+y)^2 = x^2 + 2xy + y^2.
3
Divide the numerator by the denominator to simplify the expression.
\frac{30 + 12\sqrt{6}}{6} = 5 + 2\sqrt{6}
Both integer and radical coefficients are divisible by 6.
4
Equate 5+265 + 2\sqrt{6} to a+b6a + b\sqrt{6} and calculate a+ba + b.
a = 5, b = 2, so a + b = 5 + 2 = 7.
Matching corresponding rational and irrational components.

Key Concept

Rationalization of Binomial Denominators with Surds
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Topics

Surds and Rationalization of Denominators

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