Binomial Theorem
Grade 11
Question:
<p>The sum of the rational terms in the binomial expansion of <span class="math-tex">\(\left(2^{\frac{1}{2}}+3^{\frac{1}{5}}\right)^{10}\)</span> is:</p>
<p style="display:inline">9</p>
<p style="display:inline">25</p>
<p style="display:inline">32</p>
<p style="display:inline">41</p>
Step-by-Step Solution
Key Concept: Rational terms occur in a binomial expansion when the exponents of all radical bases in the general term are integers simultaneously.
<p><span class="math-tex">$(2^{\frac 12} + 3^{\frac 15})^{10}$</span> = <sup>10</sup>C<sub>0</sub><span class="math-tex">$(2^{\frac 12})^{10}$</span> + <sup>10</sup>C<sub>1</sub><span class="math-tex">$(2^{\frac 12})^{9}(3^{\frac 15})$</span> + ... + <sup>10</sup>C<sub>10</sub><span class="math-tex">$(3^{\frac 15})^{10}$</span><br />
There are only two rational terms - first term and last term.<br />
Now sum of two rational terms<br />
= (2)<sup>5</sup> + (3)<sup>2</sup> = 32 + 9 = 41</p>
Correct Answer: D