The original expression we are given is (3^(x^7))/(27^x). To rewrite this expression in the form 3^u, we start by simplifying the expression first.
First, it's important to remember an essential rule of exponentiation: a^(m/n) equals the nth root of (a^m). In simpler terms, you divide the exponent by the root. Here, we can use this rule to simplify 27 as 3^3. This is because 27 is the cube of 3.
After this simplification, our expression now looks like this:
(3^(x^7)) / ((3^3)^x)
Next, we recall another exponentiation rule: (a^m)^n equals a^(m*n). We apply this rule to simplify (3^3)^x into 3^(3x).
Our new expression now becomes:
(3^(x^7)) / (3^(3x))
Now, when we divide with the same base, we subtract the exponents according to the rules of exponents. Thus, we subtract the exponent 3x from x^7.
Our final expression, therefore, simplifies to:
3^(x^7 - 3x)
So, we have successfully rewritten the original expression in the form 3^u. In this case, u, our algebraic expression, is (x^7 - 3x).
In conclusion, the expression (3^x^7) / (27^x) rewritten in the form 3^u is 3^(x^7 - 3x).