What is the simplified form of the surd expression \frac{4}{\sqrt{5} - 1} after rationaliz… — JAMB UTME

What is the simplified form of the surd expression $\frac{4}{\sqrt{5} - 1}$ after rationalizing the denominator?

1 + \sqrt{5}

The expression

1 + \sqrt{5}

is correct because multiplying the numerator and denominator by the conjugate

\sqrt{5} + 1

converts the denominator to

(\sqrt{5})^2 - 1^2 = 4

. Canceling the common factor of

4

in numerator and denominator simplifies the expression completely to

1 + \sqrt{5}

Identify the conjugate of the denominator

The conjugate of

\sqrt{5} - 1

\sqrt{5} + 1

Multiplying a binomial surd by its conjugate eliminates the radical in the denominator using the difference of two squares identity

(a-b)(a+b) = a^2 - b^2

Multiply both the numerator and the denominator by the conjugate

\frac{4(\sqrt{5} + 1)}{(\sqrt{5} - 1)(\sqrt{5} + 1)} = \frac{4(\sqrt{5} + 1)}{(\sqrt{5})^2 - (1)^2}

Multiplying both numerator and denominator by the same expression preserves the value of the fraction.

Simplify the denominator and evaluate the fraction

\frac{4(\sqrt{5} + 1)}{5 - 1} = \frac{4(\sqrt{5} + 1)}{4} = 1 + \sqrt{5}

Dividing the numerator by

4

cancels out the factor of

4

Rationalization of Binomial Denominators

Question