Simplifiy radicals

Question

I took an algebra exam today and came across this problem

Simplify: $ 5\sqrt{2t} - 7\sqrt{4t} + 10\sqrt{2t} $

A. $\sqrt{t}$

B. $\sqrt{2}$

C. $\sqrt{2t}$

D. $15\sqrt{2t} - 7\sqrt{4t} $

Two different approaches give two different results?

1. I did this, add like terms $ 5\sqrt{2t} +10\sqrt{2t} = 15\sqrt{2t} $

   $ 5\sqrt{2t} - 7\sqrt{4t} + 10\sqrt{2t} $

   $ 15\sqrt{2t} - 7\sqrt{4t} $ last term could be simplify to $- 14\sqrt{t} $ but it isn't an option in the test.

2. Another student did this, converting radicals to exponencials first

   $ 5\sqrt{2t} - 7\sqrt{4t} + 10\sqrt{2t} $

   $ 5(2t)^{1/2} - 7(4t)^{1/2} + 10(2t)^{1/2} $

   $ 10t^{1/2} - 28t^{1/2} + 20t^{1/2} $ I think he messed up here

   $ 2t^{1/2} = \sqrt{2t}$ also shouldn't this be $ 2t^{1/2} = 2\sqrt{t}$ ?

I chose D as the answer and he chose C, I failed that question and he didn't, am I wrong? Explain please.

Answer 1

It is $D$, clearly. You can find this yourself by eliminating other 3 cases:

What you get if, say $t=0$ and if $t=1$?

Answer 2

Put $t\rightarrow t^2$ throughout, and it comes to $$ 15 \sqrt 2 t- 14 t $$ Hence option D.

Answer 3

The answer is D. The second student made the same mistake in each term of the expression. He move the constant out of the parenthesis without taking into account the exponent. For example $7*(4t)^{1/2} = 7*4^{1/2}*t^2=2*2*t^{1/2} = 14(t)^{1/2}$ it is not $28(t)^(1/2$)