SOLUTION: Two years ago the ages of an oak tree and that of a younger oak tree were both perfect squares. Two years from now both of their ages will be perfect cubes. What is the sum, in yea

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Question 1189174: Two years ago the ages of an oak tree and that of a younger oak tree were both perfect squares. Two years from now both of their ages will be perfect cubes. What is the sum, in years, of the current ages of the two oak trees?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52787) (Show Source):
You can put this solution on YOUR website!
.

By trial and error, you can find this solution:

the younger oak three age is 6 years; the older oak three age is 123 years. The sum of their ages is 6 + 123 = 129 years. ANSWER

Answer by greenestamps(13200) (Show Source):
You can put this solution on YOUR website!

The difference between two years from now and two years ago is 4 years. Since the ages two years from now are perfect cubes and the ages two years ago are perfect squares, what you need to find is perfect cubes that are 4 greater than a perfect square.

So look at perfect cubes, subtract 4, and see if the square root of the result is an integer.
n n^3 n^3-4 sqrt(n^3-4) --------------------------- 1 1 -3 --- 2 8 4 2 3 27 23 x 4 64 60 x 5 125 121 11

That gives us the ages of the two trees as 6 and 123.

ANSWER: The sum of the ages of the two oak trees is 6+123=129 years.

HELPFUL NOTE: Similar problems can be solved using an excel file or a graphing calculator. On the spreadsheet or the graphing calculator, define a function as

and look for integer values of the function