Harvard University Aptitude Test Tricks || Algebra Problem || 99% Failed Entrance Exam || Find x-y=?

[@RyanLewis-Johnson-wq6xs]

4^x-4^y=24 2^(x+y)=35 x-y=Log[2,1.4]=Log[2,7/5]

[@mrinaldas9614]

We can start with eqn2. 35 = 7×5 or 35×1(neglected as2^35 is very large no. Negative factors are also neglected. ) => 2^x=7 2^y=5 Rest as shown. As simple as this.

[@sundareshvenugopal6575]

x = (log(10) / log(2)) - 1, y = (log(14) / log(2)) - 1.

[@michaeldoerr5810]

The answer is aimply (a,b)=(7,5) which correspinda to (x,y) = (log_2(7), log_2(5)). I shall use that as practice. Also I owe you an apology if I have been sounding like an idiot lately. Foe the 4^11-2 problem I have corrected myself 3 or 4 times and FINAL correct answer is 4,194,302. This is just two less than (2^11) squared. I better be thinking scheduling a time where I can test myself on MOST of the problems on this channel so that I can do my own "sanity checks".

[@superacademy247]

Don't worry, learning is a process. 😎😍

[@key_board_x]

2^(x + y) = 35 2^(x) * 2^(y) = 35 → let: a = 2^(x) and let: b = 2^(y) ab = 35 b = 35/a 4^(x) - 4^(y) = 24 [2^(2)]^(x) - [2^(2)]^(y) = 24 2^(2x) - 2^(2y) = 24 [2^(x)]^(2) - [2^(y)]^(2) = 24 → recall: a = 2^(x) and recall: b = 2^(y) a² - b² = 24 → recall: b = 35/a a² - (35/a)² = 24 (a⁴ - 1225)/a² = 24 a⁴ - 1225 = 24a² a⁴ - 24a² - 1225 = 0 → let: A = a² A² - 24A - 1225 = 0 Δ = (- 24)² - (4 * - 1225) = 74² A = (24 ± 74)/2 → we keep only the positive value because: A = a² A = 49 a = ± 7 → as a = 2^(x), it's an exponential, so we keep only the positive value because: A = a² a = 7 → recall: a = 2^(x) 2^(x) = 7 Ln[2^(x)] = 7 x.Ln(2) = Ln(7) → x = Ln(7)/Ln(2) Recall: b = 35/a = 35/7 b = 5 → recall: b = 2^(y) 2^(y) = 5 Ln[2^(y)] = Ln(5) y.Ln(2) = Ln(5) → y = Ln(5)/Ln(2) x - y = [Ln(7)/Ln(2)] - [Ln(5)/Ln(2)] x - y = [Ln(7) - Ln(5)] / Ln(2) → you know that: Ln(a) - Ln(b) = Ln(a/b) x - y = Ln(7/5) / Ln(2)

[@superacademy247]

Nice work

[@RealQinnMalloryu4]

(24/4^x=6Log^y24/4=6){6 ➖ 6}=0 {0+0 ➖}=1 (xy ➖ 1xy+1).{ 2*17.5}=35.0 5^7.0^0 5^7 (xy ➖ 7xy+5).

[@walterwen2975]

Harvard University Aptitude Test Tricks: 4ˣ - 4ʸ = 24, 2ˣ⁺ʸ = 35; x - y =? 4ˣ > 4ʸ > 24, x, y ϵ R; 2ˣ⁺ʸ = (2ˣ)(2ʸ) = 35 = (7)(5), 7, 5 are prime numbers 2ˣ = 7, 2ʸ = 5; 4ˣ - 4ʸ = (2ˣ)² - (2ʸ)² = 7² - 5² = 49 - 25 = 24; Confirmed 2ˣ/2ʸ = 2ˣ⁻ʸ = 7/5, x - y = log₂(7/5) = 0.485 The calculation was achieved on a smartphone with a standard calculator app Answer check: 2ˣ = 7, 2ʸ = 5: 4ˣ - 4ʸ = 24, 2ˣ⁺ʸ = 35; Confirmed as shown Final answer: x - y = log₂(7/5) = 0.485

[@superacademy247]

Great 👌 solution

[@irenehartlmayr8369]

Can you possibly have these calculations done by people who pronounce English properly ?..

[@superacademy247]

I'll consider using AI voiceover

[@irenehartlmayr8369]

Its B and NOT P !!! For goodness sake !!!!

[@superacademy247]

@irinehartlmayr8369 you're very harsh but it's alright

[@mgdacosta2]

​@@superacademy247but why do you refuse to pronounce B properly? How do you pronounce P? It is very annoying to English speskers.

[@VictorPensioner]

*** 4^x - 4^y = 24 2^(x + y) = 35 or (1) 2^(2x) - 2^(2y) = 24 (2) 2^(x + y) = 35 (1) : (2) is 2^(2x - (x + y)) - 2^(2y - (x + y)) = 24/35 or 2^(x - y) - 2^(y - x) = 24/35 Let z = 2^(x - y) (z > 0) Then z - 1/z = 24/35 or z² - (24/35) * z - 1 = 0 or 35z² - 24z - 35 = 0 z₁,₂ = [24 ± √(24² + 4 * 35²)] / 70 = (24 ± 2√(6*24 + 35²)) / 70 = (24 ± 2√(144 + 1225))/ 70 = (24 ± 2√(1369))/ 70 = (24 ± 2*37)/ 70 = (24 ± 74)/ 70 So z = 98/70 = 7/5 (z > 0) Recall 2^(x - y) = 7/5 => x - y = log₂(7/5) Threfore x - y = log₂(7/5) = log₂7 - log₂5

[@superacademy247]

Awesome 😎💯👍

[@BruceLee-io9by]

2^x=35/2^x, replace in the equation and you get the value of x and y.

[@RealQinnMalloryu4]

(24/4^xLog^y^24/4)={6 ➖ 6}= {0+0 ➖}=1 (xy ➖ 1xy+1). 2^17.5=35.0 5^7.0^0 5^7 (xy ➖ 7xy+5).

[@prollysine]

x=ln7/ln2 , y=ln5/ln2 , test , 4^x-4^y=49-25 , --> 24 , same , OK , .... , x=ln35/ln2 -y , --> , 4^(2y)+24*4^y-4^(ln35/ln2)=0 , let u=4^y , u^2+24u-4^(ln35/ln2)=0 , u=(-24 +/- 74)/2 , u= 25 , -48 , / -48 < 0 not a solu / , recall , u=4^y , 4^y=25 , y=ln25/ln4 , y=2ln5/(2ln2) , y=ln5/ln2 , x=ln35/ln2 - y , x=ln35/ln2-ln5/ln2 , x=(ln5+ln7-ln5)/ln2x , --> x=ln7/ln2 , x-y=ln7/ln2 - ln5/ln2 , x-y=(ln7-ln5)/ln2 , x-y=ln(7/5)/ln2 ,