Mathematics

Problem

Let

E=\displaystyle\frac{3\sqrt{5}+2}{6-\sqrt{5}}+\frac{4\sqrt{5}-1}{5+\sqrt{5}}

. a. (i) Rationalise each denominator separately and simplify each fraction. b. (ii) Combine the simplified fractions to obtain

E

in the form

a+b\sqrt{5}

, where

a

and

b

are rational numbers. c. (iii) Determine the numerical values of

a

and

b

. Comment on whether

E

is rational or irrational.

Intuitive explanation

The key to simplifying expressions with radicals in denominators is rationalization—multiply each fraction by a clever form of 1 that eliminates the square root below the line. For the first fraction, multiply by the conjugate of $6-\sqrt{5}$ ; for the second, factor out $\sqrt{5}$ from the denominator first, then rationalize. Once the denominators are rational, you'll be able to combine like terms and simplify to a clean answer.

Intuitive explanation

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