Final answer:
To simplify the expression, factor out the smallest power of 3, cancel common factors, evaluate the powers of 3, and divide to get the final result of 1.
Step-by-step explanation:
To simplify the expression (3m+4 - 6 · 3m+1) / (7 · 3m+2) without a calculator, we first look for a common base for all terms involving powers of 3. Noticing that each term has a power of 3, we can factor out the smallest power of 3 from the numerator.
First, rewrite the expression as:
(3m · 34 - 6 · 3m · 31) / (7 · 3m · 32)
Then factor out 3m from the numerator:
3m(34 - 6 · 31) / (7 · 3m · 32)
Now divide both numerator and denominator by 3m:
(34 - 6 · 31) / (7 · 32)
Continue to simplify by evaluating the powers of 3:
(81 - 6 · 3) / (7 · 9) = (81 - 18) / 63 = 63 / 63
The final result is: 1