First slide

Binomial theorem for positive integral Index

Question

The 10th term of ${(3 - \sqrt{\frac{17}{4} + 3 \sqrt{2}})}^{20}$ is

Moderate

A

an irrational number
B

a rational number
C

a positive integer
D

a negative integer

Solution

We have
$\frac{17}{4} + 3 \sqrt{2} = \frac{1}{4} (9 + 8 + 12 \sqrt{2}) = \frac{1}{4} (3 + 2 \sqrt{2})^{2} ∴ 3 - \sqrt{\frac{17}{4} + 3 \sqrt{2}} = 3 - \frac{1}{2} (3 + 2 \sqrt{2}) = \frac{3}{2} - \sqrt{2}$
Hence, the 10^th term of ${(3 - \sqrt{\frac{17}{4} + 3 \sqrt{2}})}^{20} = {(\frac{3}{2} - \sqrt{2})}^{20}$ is $^{20} C_{9} {(\frac{3}{2})}^{20 - 9} (- \sqrt{2})^{9}$
Which is an irrational number.

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