Harvard University Exam Question || Algebra Problem || Aptitude Simplification Test

[@VictorPensioner]

56 -24√5 = (6 - 2√5)²
Threrefore
[56 -24√5]^¼ = [(6 - 2√5)²]^¼ = √(6 - 2√5)
(6 - 2√5) = (√5 - 1)²
Threrefore
[56 -24√5]^¼ = √5 - 1

[@davidellis1929]

Trial and error: (sqrt(5)-1)^4 hits the jackpot in just a couple of minutes; just square it twice. Sometimes a little play first can lead to what would take a long time to figure out.

[@RyanLewis-Johnson-wq6xs]

Surd[56-24Sqrt[5],4]=Sqrt[5]-1 I did it in my head.

[@superacademy247]

Really?

[@RyanLewis-Johnson-wq6xs]

@@superacademy247 Yes without a doubt I do numbers like 2
101=
27318619677157413541998666579156061420147177666088128046591030596082725294498066722338505744902120368830900788923839991099564447458450075226030128555294655577015766113909738825769262480452415909200510101=101^101

[@RyanLewis-Johnson-wq6xs]

Yes really 2
101=
27318619677157413541998666579156061420147177666088128046591030596082725294498066722338505744902120368830900788923839991099564447458450075226030128555294655577015766113909738825769262480452415909200510101
=101^101

[@RyanLewis-Johnson-wq6xs]

2
101=
27318619677157413541998666579156061420147177666088128046591030596082725294498066722338505744902120368830900788923839991099564447458450075226030128555294655577015766113909738825769262480452415909200510101=101^101

[@RyanLewis-Johnson-wq6xs]

@@superacademy247 2
101=
27318619677157413541998666579156061420147177666088128046591030596082725294498066722338505744902120368830900788923839991099564447458450075226030128555294655577015766113909738825769262480452415909200510101=101^101

[@nasrullahhusnan2289]

Note that 56-24sqrt(5)>0
56-24sqrt(5)=56-2sqrt(720)
=56-2sqrt(36×20)
=[6-2sqrt(5)]²
=[sqrt(5)-1]⁴
qtrt[56-24sqrt(5)]=sqrt(5)-1

[@boguslawszostak1784]

I don't like it when someone starts with a known solution, then calculates the formula that becomes the problem, and later presents the solution "backwards" - beginning with the problem and ending with the solution.

[@jasonmark350]

^4√56-24√5
=^4√4(14-6√5)
=^4√4(3^2-2*3√5+√5^2）
=^4√2^2(3-√5）^2
=√2(3-√5）
=√(6-√5)

[@superacademy247]

One more step to reach solution