In the following equation, which variables represent rational numbers and which irrational numbers:
$u^{2} = \frac{15}{4}$

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Solution

Step 1: Solving the equation for $u$ .
$u^{2} = \frac{15}{4}$
$u^{2} - \frac{15}{4} = 0$
$u^{2} - {(\frac{\sqrt{15}}{2})}^{2} = 0$
$(u + \frac{\sqrt{15}}{2}) (u - \frac{\sqrt{15}}{2}) = 0$
$u + \frac{\sqrt{15}}{2} = 0, u - \frac{\sqrt{15}}{2} = 0$
$u = - \frac{\sqrt{15}}{2}, \frac{\sqrt{15}}{2}$
Step 2: Identifying whether $\frac{- \sqrt{15}}{2}$ and $\frac{\sqrt{15}}{2}$ are rational or irrational
$\frac{- \sqrt{15}}{2}$ and $\frac{\sqrt{15}}{2}$ are in $\frac{p}{q}$ form. But $p, q$ are not both integers.
So, $\frac{- \sqrt{15}}{2}$ and $\frac{\sqrt{15}}{2}$ are irrational numbers.
Hence, the varibale represents irrational numbers.

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