Answer:
The coefficient of t⁷ is a⁵.
Step-by-step explanation:
If we expand [tex](at-\sqrt{t} )^9[/tex], each term will has a form of [tex]\boxed{(at)^m\cdot t^{(\frac{1}{2} n)}}[/tex] where [tex]\boxed{m+n=9}[/tex].
[tex](at)^m\cdot t^{(\frac{1}{2} n)}=a^m(t^{(m+\frac{1}{2} n)})[/tex]
Therefore, for terms with t⁷, it has to fulfill these conditions:
- [tex]m+n=9[/tex] (as mentioned aboved)
- [tex]m+\frac{1}{2} n=7[/tex]
[tex]m+n=9[/tex]
[tex]m+\frac{1}{2} n=7[/tex]
------------------ (-)
[tex]\frac{1}{2} n=2[/tex]
[tex]n=4[/tex]
[tex]m+n=9[/tex]
[tex]m+4=9[/tex]
[tex]m=5[/tex]
Therefore, the term with t⁷ is [tex]a^5(t^{(5+\frac{1}{2} (4))})=a^5t^7[/tex].
The coefficient of t⁷ is a⁵.
