Seventh Root of 506623120463
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Another one from YouTube, and I want to solve it myself without the pause
to ask viewers to like, comment, subscribe and share.
So, how to find a value of x, such that x⁷=506623120463.

First, let us find two numbers between which the solution can be
found:

506623120463 is a 12-digit numbers. And we can find easily that x is
greater than 10 because 10⁷ is the digit 1 followed by 7 zeroes. 10⁷ is
an eight-digit number.
And 100⁷ is the digit 1 followed by 14 zeroes, thus 100⁷ is a 14-digit 
number.

Let us find numbers closest to our x. How about 50?
To find 50⁷, let us look at the following powers of 5:

  5, 25, 125, 625, 3125, 15625, 78125

So, 78125=5⁷, which means that 50⁷ is the 5-digit number 78125 followed by
7 zeroes, and 50⁷=781250000000>506623120463. Thus,

  x<50

Let us now look at the following powers of 4:

  4, 16, 64, 256, 1024, 4096, 16384

So, 16384=4, which means that 40⁷ is the 5-digit number 16384 followed by
7 zeroes, and 40⁷=163840000000<506623120463. Thus,

  40<x<50

Suppose x is an integer. Then because the last digit of 506623120463 is 3,
the last digit of x is 7.
Is the solution x=47?

Let us test it. First, let us look at the binomial expansion of (a-b)⁷:

  (a-b)⁷=a⁷ - 7a⁶b + 21a⁵b² - 35a⁴b³ + 35a³b⁴ - 21a²b⁵ + 7ab⁶ - b⁷

And our assumption is that a=50, b=3.
And, by looking at the following powers of 3:

  3, 9, 27, 81, 243. 729, 2187

So, 3⁷=2187. Let us start by adding and subtracting it from 506623120463:

  506623120463 = 506623120463 + 2187 - 2187
               = 506623122650 - 3⁷
               = 506623122600 + 50 - 3⁷
               = 506623122000 + 600 + 50 - 3⁷
               = 506623122000 + 50×12 + 50 - 3⁷
               = 506623122000 + 50×13 - 3⁷
               = 506623120000 + 2000 + 50×13 - 3⁷
               = 506623120000 + 50×40 + 50×13 - 3⁷
               = 506623120000 + 50×53 - 3⁷
               = 506623100000 + 20000 + 50×53 - 3⁷
               = 506623100000 + 50×400 + 50×53 - 3⁷
               = 506623100000 + 50×453 - 3⁷
               = 506623000000 + 100000 + 50×453 - 3⁷
               = 506623000000 + 50×2000 + 50×453 - 3⁷
               = 506623000000 + 50×2453 - 3⁷ 
               = 506623000000 + 50(2100+350+3) - 3⁷
               = 506623000000 + 50(7×300+7×50 + 3) - 3⁷
               = 506623000000 + 50×3 + 50×7×350 - 3⁷
               = 506620000000 + 3000000 + 50×3 + 50×7×350 - 3⁷
               = 506620000000 + 50×60000 + 50×3 + 50×7×350 - 3⁷
               = 506620000000 + 50(56000 + 4000) + 50×3 + 50×7×350 - 3⁷
               = 506620000000 + 50(7×8000 + 4000) + 50×3 + 50×7×350 - 3⁷
               = 506620000000 + 50×7×8000 + 50×4000 + 50×3 + 50×7×350 - 3⁷
               = 506620000000 + 50×7×8000 + 50(3500 + 500) + 50×3 + 50×7×350
                 - 3⁷
               = 506620000000 + 50×7×8000 + 50(7×500 + 500) + 50×3 
                 + 50×7×350 - 3⁷
               = 506620000000 + 50×7×8500 + 50×500 + 50×3 + 50×7×350 - 3⁷
               = 506620000000 + 50×7(8491+9) + 50×500 + 50×3 + 50×7×350 - 3⁷
               = 506620000000 + 50×7×8491 + 50×7×9 + 50×500 + 50×3 + 50×7×350 
                 - 3⁷
               = 506620000000 + 50×7×8491 + 50×500 + 50×3 + 50×7×359
               = 506620000000 + 50×7(8121+370) + 50×500 + 50×3 + 50×7×359 -3⁷
               = 506620000000 + 50×7×8121 + 50×7×370 + 50×500 + 50×3 + 50×7×359
                 - 3⁷
               = 506620000000 + 50×7×8121 + 50×503 + 50×7×729 - 3⁷
               = 506620000000 + 50×56847 + 50×503 + 50×7×729 - 3⁷
               = 506620000000 + 50×57350 + 50×7×3⁶ - 3⁷
               = 506620000000 + 50×50×1147 + 50×7×3⁶ - 3⁷
               = 50662000×100²+ 50²×1147 + 50×7×3⁶ - 3⁷
               = 101324000×100×50 + 50²×1147 + 50×7×3⁶ - 3⁷
               = 202648000×50² + 50²×1147 + 50×7×3⁶ - 3⁷
               = 202650000×50² - 50²×2000 + 50²×1147 + 50×7×3⁶ - 3⁷
               = 202650000×50² - 50²×853 + 50²×1147 + 50×7×3⁶ - 3⁷
               = 202655000×50² - 50²×5000 - 50²×853 + 50²×1147 + 50×7×3⁶ - 3⁷
               = 202655000×50² - 50²×5853 + 50×7×3⁶ - 3⁷
               = 202655000×50² - 50²(6000-147) + 50×7×3⁶ - 3⁷
               = 202655000×50² - 50²(6000-150+3) + 50×7×3⁶ - 3⁷
               = 202655000×50² - 50²(3×2000-3×50+3) + 50×7×3⁶ - 3⁷
               = 202655000×50² - 50²×3×1951 + 50×7×3⁶ - 3⁷
               = 202670000×50² - 50²×15000 - 50²×3×1951 + 50×7×3⁶ - 3⁷
               = 202670000×50² - 50²×3×5000 - 50²×3×1951 + 50×7×3⁶ - 3⁷
               = 202670000×50² - 50²×3×6951 + 50×7×3⁶ - 3⁷
               = 202670000×50² - 50²×3×3×2317 + 50×7×3⁶ - 3⁷
               = 202670000×50² - 50²×3×3(2100+210+7) + 50×7×3⁶ - 3⁷
               = 202670000×50² - 50²×3×3(7×300+7×3+7) + 50×7×3⁶ - 3⁷
               = 202670000×50² - 50²×3×21×331 + 50×7×3⁶ - 3⁷
               = 202670000×50² - 21×50²×993 + 50×7×3⁶ - 3⁷
               = 202670000×50² - 21×50²(750 + 243) + 50×7×3⁶ - 3⁷
               = 202670000×50² - 21×50²×750 - 21×50²×243 + 50×7×3⁶ - 3⁷
               = 202670000×50² - 21×50²×750 - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 202670000×50² - 21×50²×50×15 - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 202670000×50² - 21×50³×15 - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 202670000×50² - 7×3×50³×15 - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 202670000×50² - 7×3×3×50³×5 - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 202670000×50² - 35×50³×9 - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 4053400×50³ - 35×50³×9 - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 4049900×50³ + 3500×50³ - 35×50³×9 - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 4049900×50³ + 35×50³×100 - 35×50³×9 - 21×50²×3⁵ + 50×7×3⁶ - 3⁷ 
               = 4049900×50³ + 35×50³×91 - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 4049900×50³ + 35×50³×10 + 35×50³×81 - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 4049900×50³ + 35×50³×10 + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 4049900×50³ + 7×50³×50 + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 4049900×50³ + 7×50⁴ + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 80998×50⁴ + 7×50⁴ + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 81000×50⁴ - 2×50⁴ + 7×50⁴ + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ 
                 - 3⁷
               = 81000×50⁴ + 5×50⁴ + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 81500×50⁴ - 500×50⁴ + 5×50⁴ + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶
                 - 3⁷
               = 81500×50⁴ - 495×50⁴ + 5×50⁴ + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶
                 - 3⁷
               = 81500×50⁴ - 495×50⁴ + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 81500×50⁴ - 11×45×50⁴ + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 81500×50⁴ - 11×5×9×50⁴ + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 81500×50⁴ - 55×50⁴×3² + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 81680×50⁴ - 180×50⁴ - 55×50⁴×3² + 35×50³×3⁴ - 21×50²×3⁵ 
                 + 50×7×3⁶ - 3⁷
               = 81680×50⁴ - 20×50⁴ - 55×50⁴×3² + 35×50³×3⁴ - 21×50²×3⁵
                 + 50×7×3⁶ - 3⁷
               = 81680×50⁴ - 75×50⁴×3² + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 81680×50⁴ - 3×25×50⁴×3² + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 81680×50⁴ - 25×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 81950×50⁴ - 270×50⁴ - 25×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵ 
                 + 50×7×3⁶ - 3⁷
               = 81950×50⁴ - 10×50⁴×3³ - 25×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵
                 + 50×7×3⁶ - 3⁷
               = 81950×50⁴ - 35×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 1639×50⁵ - 35×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶ - 3⁷
               = 1600×50⁵ + 39×50⁵ - 35×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶
                  - 3⁷
               = 1550×50⁵ + 89×50⁵ - 35×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵ + 50×7×3⁶
                  - 3⁷
               = 1450×50⁵ + 189×50⁵ - 35×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵ 
                 + 50×7×3⁶ - 3⁷
               = 1450×50⁵ + 21×9×50⁵ - 35×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵ 
                 + 50×7×3⁶ - 3⁷
               = 1450×50⁵ + 21×50⁵×3² - 35×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵
                 + 50×7×3⁶ - 3⁷
               = 29×50⁶ + 21×50⁵×3² - 35×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵
                 + 50×7×3⁶ - 3⁷
               = (50-21)×5⁶ + 21×50⁵×3² - 35×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵
                 + 50×7×3⁶ - 3⁷
               = 50×50⁶ - 21×5⁶ + 21×50⁵×3² - 35×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵
                 + 50×7×3⁶ - 3⁷
               = 50⁷ - 21×5⁶ + 21×50⁵×3² - 35×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵
                 + 50×7×3⁶ - 3⁷
               = 50⁷ - 7×5⁶×3 + 21×50⁵×3² - 35×50⁴×3³ + 35×50³×3⁴ - 21×50²×3⁵
                 + 50×7×3⁶ - 3⁷
               =(50 - 3)⁷
               = 47⁷ 

Thus,
  x=47