Math Olympiad Tricks | Nice Exponent Math Simplification | Algebra Problem

hello and welcome how to simplify this expression that is 2^ 15 + 2^ 12 + 2 to^ 9 + 2 to^ 6 + 2 to^ 3 simplify and that calculators are not allowed solution now we have 2 to^ 15 + 2 to^ 12 + 2^ 9 + 2^ 6 + 2^ 3 now let us rewrite this equation so this one equals to 2 2^ 3 * 5 + 2^ 3 * 4 + 2^ 3 * 3 + 2 to^ 3 * 2 + 2^ 3 * 1 so we have changed the form of these terms of this expression so because 3 * 5 you get 15 3 * 4 you get 12 3 * 3 you get 9 3 * 2 you get 6 3 * 1 get 3 now next apply this rule that a to^ m * n is the same as a to power M the of this ra to power n by the application of this by the appliation of this rule then the of thises two so next we have 2^ 3 the of this to^ 5 plus 2 to^ 3 the all of this to^ 4 plus 2^ 3 the all of this ra to power 3 plus 2 to^ 3 the of this to power 2 plus 2 to power three of this ra to^ one now next we know that 2 to power 3 is the same as 8 therefore we have 8 to^ 5 + 8^ 4 + 8 power 3 + 8 to power power 2 + 8 is to power 1 now here let us change the form of power five and power three so next this one becomes 8 to^ 4 + 1 because 4 + 1 you get 5+ 8^ 4 + 8^ 2 + 1 so we have 2 + 1 + 8^ 2 + 8 let us apply this rule that let us apply this rule here that a^ M plus n is the same as a to power M * a^ n so we going to apply this rule here now next let us use this rule on this term and this one so this one becomes 8^ 4 * 8^ 1 + 8^ 4 + 8^ 2 * 8 to^ 1 + 8 to^ 2 + 8 as you can see between these two terms first two terms a to^ 4 is common so let us pull it out so by factorizing we have 8 to^ 4 into 8 plus + one we remain with one here so + one and the next two terms 8 squ is common so we have 8 squ by factorizing into 8 + 1 + 8 now here next this one becomes 8 + 1 you get 9 so we write 9 * 8^ 4 + 8 + 1 is 9 so 9 * 8^ 2 + 8 now next as you can see 9 is common between the two terms so this one becomes 9 into 8^ 4 + 8 8 s + 8 now this two can be written as now this one here can be written as so next this one becomes 9 into this one is the same as 8 squ all of this squ + 8 s + 8 now here we use this rule that a to power and m s is the same as a^ M * a^ m very powerful Rule now by the application of this rule therefore this one becomes this one equals to 9 into 8 2 * 8 2 + 8 s + 8 now here let us Factor the innermost because 8 squ is common let us take it out so we have 9 * now if we factorize the inmost we have 8 2 into 8 2 + 1 + 8 so here we have 9 so this one becomes 8 squ is 64 so 9 into 64 into 64 + 1 this is 64 + 1 + 8 next we have 9 into one becom 64 times this one becomes 65 + 8 64 is the same as 60 + 4 and 65 is the same as 60 + 5 now let us put in the new forms so we have 9 into 60 + 4 into 60 + 5 + 8 now here let X be equals to 60 Now by putting in X there we have we have 9 into x + 4 into x + 5 + 8 now let us expand these two factors so we have 9 into this one becomes X2 + X2 + 9 x + 20 + 8 now let us mply by 9 through out of this so we have 9 x 2 + 81 x + 9 * 20 you get 18 + 8 now let us put Let Us put 6 for X so next we have 9 * 60 2 + 81 * 60 plus this plus this we get 188 now this one becomes 81 * 60 is the same as 810 * 6 let us put this new formul so next and 60 squ is the same as 3600 so we have 9 * 3600 plus 810 * 6 + 188 now here let us multiply 9 * 0 you get 0 so you write 0 9 * 0 you get 0 you write 0 9 * 6 you get 54 so you you put four you five 9 * 3 get 27 + 5 32 so you get 32 here plus 6 * 0 you get 0 6 * 1 you get 6 6 * 8 you get 48 so you have that value here +8 8 now here let us have the sum of this these three terms so we have 32 400 plus 48 60 so we are adding so we have 06 here we have two we carry one so we have seven we have three now let us add 188 now as you can see so we have eight here we have four we call one so we have four we have seven we have three so this is the value this is the value we have obtained by addition so which means that this one equals to 37 4 48 as the final solution so which means that 2 to^ 15 + 2 to^ 12 + 2 2 power 9 + 2 2^ 6 + 2 to^ 3 = to 37 448 and that is it thank you for watching subscribe to my channel and turn on the notification B to get new updates when I upload new videos don't forget to smash the like button share and comment below to make this video reach many people

Channel: Super Academy

Views: 4,272

Keywords: exponents, algebra, simplify, olympiad, math, exponent, tricks

Id: sNeeaaZ9gsQ

Length: 11min 25sec (685 seconds)

Published: Thu Mar 28 2024