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# help

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87
5

what is the constant term in the expansion of $$\left(\sqrt{x}+\dfrac5x\right)^{9}$$?

Jan 6, 2020

### 5+0 Answers

#1
+1

The constant term is C(9,2)*5^4 = 22500.

Jan 6, 2020
#2
+1

why? pretty sure that's incorrect

Guest Jan 6, 2020
#3
+21703
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WolframAlpha.com  says 10500    but I do not know how to get there !     Let's see what Melody has to say.....

Jan 7, 2020
edited by Guest  Jan 7, 2020
#4
+108624
+2

the general term is

$$T_n=\binom {9}{n} \left(\frac{5}{x}\right)^n(x^{0.5})^{9-n}\\ T_n=\binom {9}{n} (5^n)(x^{-n})(x^{4.5-0.5n})\\ T_n=\binom {9}{n} (5^n)(x^{4.5-1.5n})\\ \text{for the constant term} \quad \\ 4.5-1.5n=0\\ 9-3n=0\\ 3n=9\\ n=3\\ T_3=\binom {9}{3} (5^3)\\ T_3=84 *125\\ T_3=10500$$

You need to check this.

Coding:

T_n=\binom {9}{n} \left(\frac{5}{x}\right)^n(x^{0.5})^{9-n}\\
T_n=\binom {9}{n} (5^n)(x^{-n})(x^{4.5-0.5n})\\
T_n=\binom {9}{n} (5^n)(x^{4.5-1.5n})\\
\text{for the constant term}  \quad \\
4.5-1.5n=0\\
9-3n=0\\
3n=9\\
n=3\\
T_3=\binom {9}{3} (5^3)\\
T_3=84 *125\\
T_3=10500

Jan 7, 2020
#5
+109061
+1

( √x  +   5/x)9

The constant term is

C(9, 6) (√x)6 (5/x)3    =

84 ( x)* 125/ (x3)    =

10500

Jan 7, 2020