Exponent Problem. Verify my solution.

Question

Is this solution correct ? please click here to see the question & my answer Sorry for the nasty handwriting (in image). $$\sqrt{a}(2a^2-4/a)\Rightarrow a^{1/2}(2a^2-4a^{-1})\Rightarrow 2(a^{5/2}-2a^{-1/2})\Rightarrow 2\sqrt{a^5-2/a}$$ $$\Rightarrow 2\sqrt{(a^6-2)/a}\Rightarrow 2(a^3/\sqrt{a}-2/\sqrt{a})\Rightarrow 2(6-2/\sqrt{a})$$

Thank you so much! :)

Answer 1

Actually, following up on my comment, I realized in general a good bit is wrong. The square root operation does not "distribute" over addition and subtraction, i.e.

$$\sqrt{x+y} \ne \sqrt{x} + \sqrt y$$

Similarly, in general for powers,

$$(x+y)^a \ne x^a + y^a$$

You use this assumption several times in your solution. For example, you claim

$$a^{5/2} - 2a^{-1/2} = \sqrt{a^5 - 2a^{-1}}$$

when in reality you can only say

$$a^{5/2} - 2a^{-1/2} = \sqrt{a^5} -2 \cdot \sqrt{a^{-1}}$$

Answer 2

It is by expanding $$2a^{2+\frac{1}{2}}-4a^{-1}\cdot a^{1/2}$$ and by simplifying $$2a^{5/2}-4a^{-1/2}$$