Evaluate $(\sqrt{3})^{11}$ (Evaluate Square Root $3$ to the power of $11$)

Question

I know the answer is $243 \sqrt 3$ but in my maths book they got $(\sqrt 3) (\sqrt 3) (\sqrt 3) (\sqrt 3) (\sqrt 3) (\sqrt 3) (\sqrt 3) (\sqrt 3) (\sqrt 3) (\sqrt 3) (\sqrt 3) (\sqrt 3)$ but then they only took the first $5$ out of the $11$ $(\sqrt 3)$s and then got $(3) (3) (3) (3) (3) (3) \sqrt 3$ and then got $3^5 \sqrt 3$

Why are they only using to the first $5$ $(\sqrt 3)$s? What happened to the other $6$ and how did the answer $243 \sqrt3$ arise? Thank you

Answer 1

What they are doing is grouping the $\sqrt 3$'s in pairs. There are $11\ \sqrt 3$s, so five pairs and one left over. Each pair becomes $\sqrt 3 \cdot \sqrt 3=3$, so they get five factors of $3$ and the left over $\sqrt 3$

Answer 2

Hint: $$3^5=243$$ and this is your answer.

Answer 3

$$ \sqrt{3}^{11} = \sqrt{3} \cdot \sqrt{3}^{10} = \sqrt{3} \cdot \left(\sqrt{3}^2\right)^5 = \sqrt{3} \cdot 3^5 = 243\sqrt{3}. $$