0. question analysis: this is a calculator not allowed question. There must be 1 or more ways can extremely simplify the qustion. 1.x^6 -1 = (x-1)(x^5+x^4+x^3+x^2+x+1) => (x^5+x^4+x^3+x^2+x)= [(x^6 -1)/(x-1)]-1, let x=27. 2. Ans= [(27^6 -1)/(27-1)]-1=[(3^18 -1)/26]-1=[(9^9 -1)/26]-1, this is simple enough to calculate directly. 3.9^9 can calculate directly or solved by (x-1)^n = C(n,0)x^n - C(n,1)x^n-1 + C(n,2)x^n-2 - C(n,3)x^n-3......，x=10, n=9
The expression is the same as 27^6 - 1/27-1...this is (27^3 -1) (27^3+1)/27-1. The numerator can further be factories using difference and sum of cubes formula
^=read as to the power *=read as square root As per question 27^5 +27^4+27^3+27^2+27^1 Add 1 to the above statement 1+27+27^2+27^3+27^4+27^5 =(1+27)+(27^2+27^3)+(27^4+27^5) =(1+27)+27^2(1+27)+27^2(1+27) =(1+27){1+27^2+27^4} =28[1+(2×27^2)+(27^2)^2}-(27^2)] =28[(27^2+1)^2-(27^2)] =28{(729+1)^2- 729} =28{(730^2)-729} =28{532900-729} =28×532171 =14900788 Hence the conclusion will be 14900788-1=14900787..(because 1 is added to the statement earlier)
The expression is the same as 111110 base 27, which is 14,900,787 base 10 as any elementary student should know. How is this even a question??? 😂 jk, happy holidays and thanks for a great vid!
Let equal.= K and multiple both side with (27-1). Then use formula, we could convey it as following. 27^6-1^6=26K (27^2)^3-1^3=26K 729^3-1^3=26K (729-1)(729^2+729+1^2)=26K 728(729*730+1)=26K 28(532171)=K 14900788=K
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Respected Sir, Good evening.. Nicely solved
I appreciate you watching! 🤩🥰💕Thanks for your support 🙏💕💯
0. question analysis: this is a calculator not allowed question. There must be 1 or more ways can extremely simplify the qustion.
1.x^6 -1 = (x-1)(x^5+x^4+x^3+x^2+x+1) => (x^5+x^4+x^3+x^2+x)= [(x^6 -1)/(x-1)]-1, let x=27.
2. Ans= [(27^6 -1)/(27-1)]-1=[(3^18 -1)/26]-1=[(9^9 -1)/26]-1, this is simple enough to calculate directly.
3.9^9 can calculate directly or solved by (x-1)^n = C(n,0)x^n - C(n,1)x^n-1 + C(n,2)x^n-2 - C(n,3)x^n-3......，x=10, n=9
Thanks for sharing your method. This is a nice alternative to the approach in the video! 💕🥰💕💪
The expression is the same as 27^6 - 1/27-1...this is (27^3 -1) (27^3+1)/27-1. The numerator can further be factories using difference and sum of cubes formula
Thanks for sharing your alternative approach to solving this problem! 🔥🔥✅💕
Standard elementary math form, called a geometric series.
Используем формулу суммы членов геометрической прогрессии.
b1=27
b5=27^5
q=27
S=(27^6-27)/(27-1).
Good luck!
^=read as to the power
*=read as square root
As per question
27^5 +27^4+27^3+27^2+27^1
Add 1 to the above statement
1+27+27^2+27^3+27^4+27^5
=(1+27)+(27^2+27^3)+(27^4+27^5)
=(1+27)+27^2(1+27)+27^2(1+27)
=(1+27){1+27^2+27^4}
=28[1+(2×27^2)+(27^2)^2}-(27^2)]
=28[(27^2+1)^2-(27^2)]
=28{(729+1)^2- 729}
=28{(730^2)-729}
=28{532900-729}
=28×532171
=14900788
Hence the conclusion will be
14900788-1=14900787..(because 1 is added to the statement earlier)
Easier just to calculate conventionally. The method used complicates rather than simplifies.
this problem and the solution is available in MECHANICAL ENGINEERS HAND BOOK
I appreciate you sharing your insights! 💯🤩🙏✅
The expression is the same as 111110 base 27, which is 14,900,787 base 10 as any elementary student should know. How is this even a question??? 😂 jk, happy holidays and thanks for a great vid!
Let equal.= K and multiple both side with (27-1). Then use formula, we could convey it as following.
27^6-1^6=26K
(27^2)^3-1^3=26K
729^3-1^3=26K
(729-1)(729^2+729+1^2)=26K
728(729*730+1)=26K
28(532171)=K
14900788=K
27^2*(27^3+27^2)+(27^3+27^2)+27=(27^3+27^2)(27^2+1)+27
=(19683+729)(729+1)+27
=20412*730+27=14,900,787