If the expression \frac{3 - \sqrt{5}}{3 + \sqrt{5}} is simplified and written in the form … — JAMB UTME

If the expression $\frac{3 - \sqrt{5}}{3 + \sqrt{5}}$ is simplified and written in the form $a + b\sqrt{5}$ , where $a$ and $b$ are rational numbers, what is the numerical value of $a + b$ ?

Answer: 2

Answer

The numerical value of

a + b

2

To express

\frac{3 - \sqrt{5}}{3 + \sqrt{5}}

in the standard form

a + b\sqrt{5}

, multiply both the numerator and denominator by the conjugate of the denominator, which is

(3 - \sqrt{5})

. The numerator expands to

(3 - \sqrt{5})^2 = 9 - 6\sqrt{5} + 5 = 14 - 6\sqrt{5}

. The denominator becomes

3^2 - (\sqrt{5})^2 = 9 - 5 = 4

. Dividing gives

\frac{14}{4} - \frac{6}{4}\sqrt{5} = \frac{7}{2} - \frac{3}{2}\sqrt{5}

. Hence,

a = \frac{7}{2}

and

b = -\frac{3}{2}

, making

a + b = \frac{7}{2} - \frac{3}{2} = \frac{4}{2} = 2

Step-by-Step Solution

Multiply numerator and denominator by the conjugate of the denominator

\frac{(3 - \sqrt{5})(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})}

To eliminate the surd from the denominator.

Expand both the numerator and the denominator

\frac{14 - 6\sqrt{5}}{4}

Using

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

for the numerator and difference of two squares

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

for the denominator.

Separate into rational component and radical coefficient

\frac{7}{2} - \frac{3}{2}\sqrt{5}

Simplifying fractions by dividing numerator and denominator by their greatest common divisor.

Calculate the sum

a + b

\frac{7}{2} - \frac{3}{2} = 2

Comparing

\frac{7}{2} - \frac{3}{2}\sqrt{5}

with

a + b\sqrt{5}

yields

a = \frac{7}{2}

and

b = -\frac{3}{2}

Key Concept

Rationalization of Binomial Denominators using Conjugates

Estimated Time:1m 30s

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