Please tell me what I did wrong.
This is a question from my review packet.
The answer says it should be 2 with exponent 3/2
My attempt gives me this:
Your mistakes are in the third line. You can't just take the exponents out of the square root like that!!
Note that:
$\displaystyle 2^{5/2} = 2^2 \times 2^{1/2} = 4 \sqrt{2} \, $ and $\displaystyle \, 2^{3/2} = 2^1 \times 2^{1/2} = 2 \sqrt{2}$.
So your problem becomes $\displaystyle 4 \sqrt{2} - 2 \sqrt{2} = 2 \sqrt{2}$.
Now note that $\displaystyle 2 \sqrt{2} = 2^{3/2}$.
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Alternatively, note that $\displaystyle 2^{5/2} = 2^1 \times 2^{3/2} = 2 \times 2^{3/2}$.
So your problem becomes $\displaystyle 2 \times 2^{3/2} - 2^{3/2} = 2^{3/2}$.