Real numbers
Grade Class 10
Question:
<p>The number <span class="math-tex">\({(\sqrt 3 + \sqrt 5 )^2}\)</span> is</p>
<p style="display:inline">an integer</p>
<p style="display:inline">an irrational number</p>
<p style="display:inline">not a real number</p>
<p style="display:inline">a rational number</p>
Step-by-Step Solution
Key Concept: Expand the expression using the identity $(a+b)^2 = a^2 + b^2 + 2ab$ and determine if the resulting sum contains any remaining irrational terms.
<p><span class="math-tex">\({\left( {\sqrt 3 + \sqrt 5 } \right)^2}\)</span> = <span class="math-tex">\({\left( {\sqrt 3 } \right)^2} + {\left( {\sqrt 5 } \right)^2} + 2 \times \sqrt 3 \times \sqrt 5 \)</span><br />
= <span class="math-tex">\(3 + 5 + 2\sqrt {15} \)</span><br />
= <span class="math-tex">\(8 + 2\sqrt {15} \)</span><br />
Here, <span class="math-tex">\(\sqrt {15} = \sqrt 3 \times \sqrt 5 \)</span><br />
Since <span class="math-tex">\(\sqrt 3 \)</span> and <span class="math-tex">\(\sqrt 5 \)</span> both are an irrational number. Therefore, <span class="math-tex">\({\left( {\sqrt 3 + \sqrt 5 } \right)^2}\)</span> is an irrational number.</p>
Correct Answer: B