State, whether the following numbers are rational or irrational.
$(5 - \sqrt{5})^{2}$

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Solution

Step: Check whether the expression is rational or irrational.

$(5 - \sqrt{5})^{2}$

$= (5^{2}) + ({\sqrt{5}}^{2}) - 2 \times 5 \times \sqrt{5}$

On further calculation we get,

$= 25 + 5 - 10 \sqrt{5}$

$= 30 - 10 \sqrt{5}$

Which is an irrational number

Therefore $(5 - \sqrt{5})^{2}$ is an irrational number.

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