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  • The number of terms which are free from fractional powers in the expansion of <img alt="\left(a^{1 / 5}+\right.\left.b^{2 / 3}\right)^{45}a \neq b \text { is }" src="https://entrancecorner.onc

The number of terms which are free from fractional powers in the expansion of \left(a^{1 / 5}+\right.\left.b^{2 / 3}\right)^{45}a \neq b \text { is }

Option: 1

9


Option: 2

15


Option: 3

4


Option: 4

None of these


Answers (1)

best_answer

The general term in the expansion of  \left(a^{1 / 5}+b^{2 / 3}\right)^{45} \text { is }

\begin{aligned} T_{r+1} & ={ }^{45} C_r\left(a^{1 / 5}\right)^{45-r}\left(b^{2 / 3}\right)^r \\ & ={ }^{45} C_r a^{9-(r/5)} b^{2 r / 3} \end{aligned}

This will be free from fractional powers if both r/5 and 2r/3 are whole numbers i.e. if r = 0, 15, 30, 45.
Hence, there are only four terms which are free from fractional powers.

Posted by

shivangi.bhatnagar

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