Lots Of 7 & Lots Of Luck

By rules of exponents , $a^{m} \div a^{n} = a^{m-n}$ and $a^{m} \times a^{n} = a^{m+n}$ .

The expression in the denominator simplifies to $\large (7\times 7)^7 = 7^7 \times 7^7 = 7^{7 + 7} = 7^{14}$ .

Whereas the expression the numerator is already simplified.

Now let's simplify the expression in question!

${ \begin{aligned} \dfrac{7^{7^7} }{ (7\times7)^7} &=& \dfrac{7^{7^7} }{ 7^{14}} \\ &=& 7^{7^7} \div 7^{14} \\ &=& \large\boxed{ 7^{7^7 -14}} . \\ \end{aligned} }$

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Footnote : But why can't we simplify $7^{7^7}$ to $(7^7)^7 = 7^{7\times7} = 7^{49}$ ? Unfortunately, this is not true because $\large a^{b^c} = a^{b\times c}$ is not an algebraic identity. You may read further in the following wiki: how are exponent towers evaluated? is $a^{b^c}$ equal to $a^{b \times c}$ ? .

$\begin{aligned} \frac{ 7^{7^{7} } }{(7\cdot7)^7} &= \frac{ 7^{7^{7}} }{7^7 \cdot 7^7} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad{\text{Use} : a^m \cdot a^n = a^{m+n} }\\&=\frac{7^{7^{7}}}{7^{14}} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad{\text{Use}: \frac{a^m}{a^n} = a^{m-n}} \\&= 7^{7^{7}-14} \space \text{or} 7^{823529} \space\space\space \square \end{aligned}$