Show that is irrational.

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Rationalisation

Question

Show that $5 + 3 \sqrt{5}$ is irrational.

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Let us assume

5 + 3 \sqrt{5}

is rational

5 + 3 \sqrt{5} = \frac{a}{b}

3 \sqrt{5} = \frac{a}{b} - 5

\sqrt{5} = \frac{a}{3 b} - \frac{5}{3}

here

\sqrt{5}

is lrrational

\frac{a}{3 b} - \frac{5}{5}

is rational

i,e which condradicts the statement

∴

5 + 3 \sqrt{5}

is irrational

Hence proved