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If $18\sqrt{18}=r\sqrt{t}$ where $r$ and $t$ are positive integers and $r>t$, which of the following could be the value of $rt$?

A. 18
B. 36
C. 108
D. 162
E. 324

Thus far I have tried to solve for $r\sqrt{t}$ and I have gotten $r^2t=5832$ . Is this so far correct?

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How did you get a decimal number for $r^2t$? When you square $18\sqrt{18}$ you get $18^3 = 5832$. –  Jim Mar 3 '13 at 0:26
    
you're absolutely right my mistake. –  Little Jon Mar 3 '13 at 0:40

2 Answers 2

up vote 4 down vote accepted

We know that $$r\sqrt t = 18\sqrt {18},$$ but we need $r>t$, so $r = t = 18$ won't work.

Recall that $$\sqrt{ab} = \sqrt a \times \sqrt b$$

And note that $\sqrt{18} = \sqrt 9 \times \sqrt 2$. So

$$18\sqrt{18} = 18 \times \sqrt{9\cdot 2} $$ $$ = 18 \times \sqrt 9 \times \sqrt 2 $$ $$ = 18 \times 3 \times \sqrt 2 $$ $$ = 54 \sqrt 2$$

So now if we let $r = 54$ and $t = 2$, then $r>t$, as desired, and $$rt = 54\times 2 = 108$$

And $\;rt = 108\;$ is on our list: as choice $\,(C)\,$!

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You could look at $54=3\cdot3\cdot3\cdot2$ and then you have $$54\sqrt{2}=3\cdot3\cdot3\cdot2\cdot\sqrt{2}.$$ Since $2 = \sqrt{4}$ we could replace $2$ with $\sqrt{4}$ to get $$54\sqrt{2}=3\cdot3\cdot3\cdot\sqrt{4}\cdot\sqrt{2}=3\cdot3\cdot3\cdot\sqrt{4 \cdot 2}=27\sqrt{8}$$ so $r=27$ and $t=8$ would work too though their product $216$ isn't listed. –  Starlight Mar 3 '13 at 1:32
    
Excellent Well done! –  Little Jon Mar 3 '13 at 19:59

Well right now you have both sides of the equation in the same form, but the problem states that $r$ is greater than $t$ so we know they cannot both equal 18. Simplify the square root on the right side of the equation to obtain:

$$ \begin{align*} 18\sqrt{18} &= 18 \sqrt{9 \cdot 2} \\ &= 18 \cdot 3 \sqrt{2} \\ &= 54 \sqrt{2} \end{align*} $$ So $r = 54$ and $t = 2$, $r \cdot t = 108$

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Very nice. Have a mistake on the left hand side, though. It isn't 18. –  vonbrand Mar 3 '13 at 0:54

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