If the expression \frac{3\sqrt{2} - 2\sqrt{3}}{3\sqrt{2} + 2\sqrt{3}} can be expressed in … — State PSC Exam

If the expression $\frac{3\sqrt{2} - 2\sqrt{3}}{3\sqrt{2} + 2\sqrt{3}}$ can be expressed in the form $a - b\sqrt{6}$ where $a$ and $b$ are rational numbers, what is the exact value of $a + b$ ?

Answer: 7

Answer

The correct value is 7.

The correct answer is derived by multiplying the numerator and denominator by the conjugate

3\sqrt{2} - 2\sqrt{3}

. This rationalizes the denominator to

6

. Expanding the numerator gives

30 - 12\sqrt{6}

. Dividing the numerator by

6

yields

5 - 2\sqrt{6}

. Setting this equal to

a - b\sqrt{6}

identifies

a = 5

and

b = 2

, giving a final sum of

7

Step-by-Step Solution

Multiply the numerator and denominator by the conjugate of the denominator,

3\sqrt{2} - 2\sqrt{3}

\frac{(3\sqrt{2} - 2\sqrt{3})^2}{(3\sqrt{2} + 2\sqrt{3})(3\sqrt{2} - 2\sqrt{3})}

This process, known as rationalizing the denominator, removes the surds from the bottom of the fraction.

Expand the numerator using the binomial square formula

(x-y)^2 = x^2 - 2xy + y^2

(3\sqrt{2})^2 - 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 18 - 12\sqrt{6} + 12 = 30 - 12\sqrt{6}

Expanding the squared binomial simplifies the top part of the fraction.

Expand the denominator using the difference of squares formula

(x+y)(x-y) = x^2 - y^2

(3\sqrt{2})^2 - (2\sqrt{3})^2 = 18 - 12 = 6

This guarantees that the denominator becomes a rational number.

Divide the terms in the numerator by the denominator.

\frac{30 - 12\sqrt{6}}{6} = 5 - 2\sqrt{6}

Simplifying the fraction allows us to match it to the given form

a - b\sqrt{6}

Equate the simplified expression to

a - b\sqrt{6}

and solve for

a + b

a = 5

b = 2

, and

a + b = 7

By direct comparison of rational and irrational parts, we determine the values of

a

and

b

to find their sum.

Key Concept

Rationalizing the denominator using conjugates and expanding binomial expressions involving surds.

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