Answer: Choice B) [tex]\sqrt{63}[/tex]
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Explanation:
Let's rewrite the given expression like so
[tex]3\sqrt{y+3}=\sqrt{3^2}\sqrt{y+3}\\\\3\sqrt{y+3}=\sqrt{9}\sqrt{y+3}\\\\3\sqrt{y+3}=\sqrt{9(y+3)}\\\\[/tex]
We can see that the radicand is a multiple of 9 since 9 is a factor.
This immediately rules out choices C and D because 75 and 84 are not multiples of 9.
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Plug in the smallest positive integer (1) for y to get
[tex]\sqrt{9(1+3)}=\sqrt{9(4)} = \sqrt{36}\\\\[/tex]
The radicand 36 is larger than the radicand for choice A (18), meaning we can rule out choice A. There's no way to get a radicand of 18 since y = 1 is the smallest we can go for.
We can however get to 63 by using y = 4
[tex]\sqrt{9(4+3)}=\sqrt{9(7)} = \sqrt{63}\\\\[/tex]
Showing that choice B is of the form [tex]3\sqrt{y+3}[/tex] where y is a positive integer, specifically when y = 4.
