these are not university-level algebra questions.
for (1):
since everything in sight is positive we can square both sides without fear.
thus from:
18√18 = r√t, we have:
(324)(18) = r
^{2}t
18
^{3} = 5832 = r
^{2}t.
clearly t < 18, or else r
^{2}t > t
^{3} > 5832. so t is some divisor of 18: 1,2,3,6,or 9.
if t = 1, r = √(5832), which is not an integer.
if t = 2, r = √(2916) = 54 <--this works ( (18)
^{3}/2 = (9)(18)
^{2}, which has square root 3*18 = 54).
if t = 3, r = √(1944), not an integer
if t = 6, r = √(972), not an integer
if t = 9, r = √(648), not an integer
(look at the prime factorization of 18 cubed)
so the only case where r and t are integers with r > t is t = 2, r = 54, hence rt = 108.
(2) 2x + 3x = 5x, the number of eggs in the basket must be a multiple of 5. 12 is not a multiple of 5.
MY FOLLOW UP QUESTION
This is for question 1. (I feel a bit sheepish for posting question 2 once I realized it
)
From Denevo's reply:
"since everything in sight is positive we can square both sides without fear.
thus from: √18 = r√t, we have:
(324)(18) = r^{2}t
18^{3} = 5832 = r^{2}t.
clearly t < 18, or else r^{2}t > t^{3} > 5832. so t is some divisor of 18: 1,2,3,6,or 9."
I think where I'm getting hung up in the explanation is the "
clearly t < 18, or else r^{2}t > t^{3} > 5832. so t is some divisor of 18: 1,2,3,6,or 9."
I understand that if t > r that the result would be > 5832. The part I'm having trouble with is how to indentify that t is in fact less than 18 specifically. How do I know it's 18 and not some other integer? In other words, by saying this, it seems like saying that means you understand that r would be = 18, thus t < 18. From the problem, all I know is that r > t, and that the product of r*sqrt(18) is equivalent to the product of r*sqrt(t), and that r doesn't equal 18 nor does t equal sqrt(18). It seems like this is also confirmed by the solution that r = 54 and t = 2.
I certainly hope the follow question I'm asking here is making sense. Also, is there a link/website or some section of math one would practice in a book/class for a problem of this nature? It's important to me to understand the "why" of it instead of memorizing "how" to do a particular problem type.
Thanks.