(1+root 3)whole square is rational or irrational

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Solution

$Let x = {(1 + \sqrt{3})}^{2} = {(1)}^{2} + {(\sqrt{3})}^{2} + 2 \times 1 \times \sqrt{3} = 1 + 3 + 2 \sqrt{3} = 4 + 2 \sqrt{3}, which is an irrational number . Hence, x = {(1 + \sqrt{3})}^{2} represents an irrational number .$

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(1+root 3)whole square is rational or irrational

Let x = 1+32=12 + 32 + 2 × 1 × 3=1 + 3 + 23=4 + 23 , which is an irrational number.Hence, x = 1+32 represents an irrational number.

$Let x = {(1 + \sqrt{3})}^{2} = {(1)}^{2} + {(\sqrt{3})}^{2} + 2 \times 1 \times \sqrt{3} = 1 + 3 + 2 \sqrt{3} = 4 + 2 \sqrt{3}, which is an irrational number . Hence, x = {(1 + \sqrt{3})}^{2} represents an irrational number .$