Wow! That is such a mouthful of an expression, but I will attempt to evaluate it anyway.
If I am not mistaken, the expression is the following:
\((\frac{62^{62}-62^{62-1}}{62^{62-1}})^{\sqrt{62^{62-1}+62^{62}-62^{62-2}}}\)
\((\frac{62^{62}-62^{62-1}}{62^{62-1}})^{\sqrt{62^{62-1}+62^{62}-62^{62-2}}}\) | I'll start with the fraction portion of this expression |
\(\frac{62^{62}-62^{62-1}}{62^{62-1}}\) | First, I will break the fraction up with the following fraction rule that states \(\frac{a\pm b}{c}=\frac{a}{c}\pm\frac{b}{c}\). |
\(\frac{62^{62}-62^{62-1}}{62^{62-1}}=\frac{62^{62}}{62^{62-1}}-\frac{62^{62-1}}{62^{62-1}}\) | Of course, \(\frac{62^{62-1}}{62^{62-1}}=1 \) because any nonzero number divided by itself is always one. |
\(\frac{62^{62}}{62^{62-1}}-\frac{62^{62-1}}{62^{62-1}}=\frac{{62^{62}}}{62^{61}}-1\) | We will utilize an exponent rule that states that \(\frac{a^b}{a^c}=a^{b-c}\). |
\(\frac{{62^{62}}}{62^{61}}=62^{62-61}=62^1=62\) | Reinsert this in for \(\frac{{62^{62}}}{62^{61}}\). |
\(62-1=61\) | Ok, we have successfully simplified from \(\frac{62^{62}-62^{62-1}}{62^{62-1}}\) to 61. Reinsert 61 for \(\frac{62^{62}-62^{62-1}}{62^{62-1}}\) in the original expression. |
\(61^{\sqrt{62^{62-1}+62^{62}-62^{62-2}}}\) | I will convert \(62^{62-1}\) into a fraction by using the converse of a fraction rule I utilized before. It is \(a^{b-c}=\frac{a^b}{a^c}\). |
\({\sqrt{62^{62-1}+62^{62}-62^{62-2}}}={\sqrt{\frac{62^{62}}{62^1}+62^{62}-62^{62-2}}}\) | Multiply \(62^{62}\) by \(\frac{62}{62}\) to create a common denominator. |
\(\frac{62^{62}}{1}*\frac{62}{62}=\frac{62*62^{62}}{62^1}\) | Insert this back into the equation, too. |
\({\sqrt{\frac{62^{62}}{62^1}+62^{62}-62^{62-2}}}={\sqrt{\frac{62^{62}}{62^1}+\frac{62*62^{62}}{62^1}-62^{62-2}}}\) | Add \(\frac{62^{62}}{62^1}+\frac{62*62^{62}}{62^1}\) together. |
\({\sqrt{\frac{62^{62}}{62^1}+\frac{62*62^{62}}{62^1}-62^{62-2}}}=\sqrt{\frac{63*62^{62}}{62^1}-62^{60}}\) | Convert \(\frac{62^{60}}{1}*\frac{62}{62}\) into a fraction over 62 by doing |
\(\frac{62^{60}}{1}*\frac{62}{62}=\frac{62^{60}*62}{62}=\frac{62^{61}}{62}\) | Replace \(62^{60}\) with \(\frac{62^{61}}{62}\). |
\(\sqrt{\frac{63*62^{62}}{62^1}-62^{60}}=\sqrt{\frac{63*62^{62}}{62^1}-\frac{62^{61}}{62}}\) | Subtract the fractions. |
\(\sqrt{\frac{63*62^{62}}{62^1}-\frac{62^{61}}{62}}=\sqrt{\frac{63*62^{62}-62^{61}}{62}}\) | "Distribute" the square root to both the numerator and denominator. |
\(\sqrt{\frac{63*62^{62}-62^{61}}{62}}=\frac{\sqrt{63*62^{62}-62^{61}}}{\sqrt{62}}\) | Rationalize the denominator by multiplying \(\frac{\sqrt{63*62^{62}-62^{61}}}{\sqrt{62}}\) by \(\frac{\sqrt{62}}{\sqrt{62}}\). |
\(\frac{\sqrt{63*62^{62}-62^{61}}}{\sqrt{62}}*\frac{\sqrt{62}}{\sqrt{62}}\) | Use the square root rule that states that \(\sqrt{x}*\sqrt{y}=\sqrt{xy}\) |
\(\frac{\sqrt{62(63*62^{62}-62^{61})}}{62}\) | Distribute the 62 to all terms. |
\(\frac{\sqrt{62(63*62^{62}-62^{61})}}{62}=\frac{\sqrt{62*63*62^{62}-62*62^{61}}}{62}\) | Simplify. |
\(\frac{\sqrt{3906*62^{62}-62^{62}}}{62}\) | Do the subtraction in the numerator. |
\(3906*62^{62}-62^{62}=3905*62^{62}\) | Reinsert that back into the expression. I will simplify \(\sqrt{3905*62^{62}}\) by using the rule that \(\sqrt{ab}=\sqrt{a}*\sqrt{b}\) |
\(\sqrt{3905}*\sqrt{62^{62}}\) | |
\(\sqrt{62^{62}}=\sqrt{62^{31}*62^{31}}=62^{31} \) | Plug this back into the original expression |
\(\frac{62^{31}\sqrt{3905}}{62}=62^{30}*\sqrt{3905}\) | Factor out the common factor of 62 |
At this point, I cannot simplify the exponent any further. I have done an impressive job of simplifying
\((\frac{62^{62}-62^{62-1}}{62^{62-1}})^{\sqrt{62^{62-1}+62^{62}-62^{62-2}}}\) into \(61^{({62^{30}*\sqrt{3905}})}\). I will now have to utilize an extremely accurate calculator for a calculation of this size. My only option is to provide you with an exponent tower; that's how huge it is!
\((\frac{62^{62}-62^{62-1}}{62^{62-1}})^{\sqrt{62^{62-1}+62^{62}-62^{62-2}}}=61^{({62^{30}\sqrt{3905}})}\approx 10^{10^{55.81927966668297}}\)