AS Maths simplify

Question

A question in my AS Exam this morning was simply, Simplify $(5\sqrt5)^3$

I tried $(5\sqrt5 \cdot 5\sqrt5)^2$ and ended with $5^4$. Is that correct?

I think it's completely wrong, but it'd be awesome if someone could point my problem out. Thank you!

EDIT: I've forgotten that the question wanted the answer writen as $5^N$

Answer 1

The problem is that $$ (5\sqrt 5)^3 = 5\sqrt 5 \cdot 5\sqrt 5 \cdot 5\sqrt 5 $$ and $$ (5\sqrt 5\cdot 5\sqrt 5)^2 = 5\sqrt 5 \cdot 5\sqrt 5 \cdot 5\sqrt 5\cdot 5\sqrt 5 $$ so you have one $5\sqrt 5$ too much. EDIT: In case you want your answer as $5^n$, note that we have $$ (5\sqrt 5)^3 = 5\sqrt 5 \cdot 5\sqrt 5 \cdot 5\sqrt 5 = 5 \cdot 5^\frac12\cdot5\cdot 5^\frac12\cdot 5^\frac12 $$ Can you apply some rules to write this like $5^n$?

Answer 2

No. $(5\sqrt 5 \cdot 5\sqrt 5)^2 = ((5 \sqrt 5)^2)^2 =(5 \sqrt 5)^4$, not $(5 \sqrt5)^3$.

You can do this: $(5 \sqrt5)^3 = 5^3 \sqrt5 ^3 = 5^4 \sqrt 5 = 625\sqrt5$

Answer 3

Let $u = 5\sqrt 5$. Your mistake was in simplifying $u^3$ to $(uu)^2$, because $(uu)^2 = (u^2)^2 = u^4$.

Answer 4

It's $(5\cdot5^{1/2})^3=(5^{3/2})^3=5^{9/2}=5^4\sqrt{5}$.

Answer 5

it is $$5\sqrt{5} \cdot 5\sqrt{5} \cdot 5\sqrt{5}=125\cdot 5\sqrt{5}=625\sqrt{5}$$ and $625\sqrt{5}=5^4\cdot 5^{1/2}=5^{9/2}$

Answer 6

$$(5\sqrt5)^3=(5^1\times5^{1/2})^3=(5^{3/2})^3=5^{(3/2) \times 3}=5^{9/2}$$