American | Junior Math Olympiad | A Nice Algebra Problem | Simplify?

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Junior Math Olympiad
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[@walterwen2975]

Junior Math Olympiad: 5¹¹ - 3² = ? 5¹¹ = (5³)(5²)⁴ = 125[(25)⁴] = 125[(25²)²] = 125(625²) 625² = (600 + 25)² = 600² + 2(600)(25) + 25² = 360625 + 30000 = 390600 + 25 125(625²) = (100 + 25)(390600 + 25) = 39060000 + 25(390600) + 2500 + 625 = 39063125 + (20 + 5)(390600) = 39063125 + 7812000 + 1953000 = 48828125 5¹¹ - 3² = 48828125 - 9 = 48828116

[@alexanderbrinkley4332]

In the time you have written all the manipulations, you could have easily raised 5 to the 11th power and subtracted 9.

[@Mal1234567]

It said to simplify, not solve, without using a calculator.

[@Mal1234567]

{9 = 5 + 4} What is this, some rare theorem we've never heard of before? LOL.

[@VictorPensioner]

5^11 = (5^11 * 2^11) / 2^11 = 10^11 / 2^11 = 1 000 000 000 00 / (2 * 1024) = 1 000 000 000 00 / 2048 = 48828125 5^11 - 3^2 = 48828125 - 9 = 48828116

[@Mal1234567]

Much of this solution requires a calculator unless you have the solution to 31 x 31 memorized, and other things like that.

[@henridegueldre1202]

Are you sure that it wasn't be best to calcultate 5^11 directly ? 10 minutes to manipulate, without strong idea, 5^11-3^2...

[@Mal1234567]

It said to simplify, not solve.

[@hn4806]

Your approach does not seem any better than direct calculation. 5**11=[(5**3)**3]x25=125**3x25 ~ 4 multiplication knowing 5**3=125 and 5**2=25. Much smaller # of multiplications with this no-brainer approach.

[@Mal1234567]

It said to simplify, not solve, without a calculator.