Question
Classify the following number as rational or irrational:$\big(\sqrt3+\sqrt5\big)$
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Answer
Let $\sqrt3+\sqrt5$ be rational. $\therefore\sqrt3+\sqrt5= \text{a},$ where aa is rational $\therefore\sqrt3=\text{a}-\sqrt5\dots(1)$ On squaring both sides of equation (1), we get$3=\big(\text{a}-\sqrt5\big)^2$
$=\text{a}^2+5-2\sqrt5\text{a}$ $\Rightarrow\sqrt5=\frac{\text{a}^2+2}{\text{2a}}$ This is impossible because right-hand side is rational, whereas the left-hand side is irrational. This is a contradiction. Hence, $\sqrt3+\sqrt5$ is irrational.
$=\text{a}^2+5-2\sqrt5\text{a}$ $\Rightarrow\sqrt5=\frac{\text{a}^2+2}{\text{2a}}$ This is impossible because right-hand side is rational, whereas the left-hand side is irrational. This is a contradiction. Hence, $\sqrt3+\sqrt5$ is irrational.
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