Solve (5x + 3 to the 4th) times 6 to the 8th equals 6 to the 9th times 3 to the 4th.

Step-by-step solution

1. 1

   Spot which factor is common to both sides. Both sides carry a power of $6$; the $3^4$ appears inside the bracket on the left but as a separate factor on the right, so it does not cancel. Only $6^8$ can be divided out.

2. 2

   Divide both sides by 6^8. $5x+3^4 = \frac{6^9\cdot3^4}{6^8} = 6^{9-8}\cdot3^4 = 6\cdot 3^4.$

3. 3

   Evaluate the small powers. $3^4 = 81$, so the equation becomes $5x+81 = 6\cdot81 = 486.$

4. 4

   Solve the linear equation. $5x = 486-81 = 405 \quad\Longrightarrow\quad x = \frac{405}{5} = 81.$ Pleasingly, $x$ comes out equal to $3^4$ itself, since $5x = 5\cdot81$ follows from $6\cdot81-81 = 5\cdot81$.

5. 5

   Verify. Left side: $(5\cdot81+81)\cdot6^8 = 486\cdot6^8 = 6\cdot81\cdot6^8 = 81\cdot6^9$. Right side: $6^9\cdot81$. Identical.

6. 6

   Note the shortcut. Recognising $486 = 6\cdot81$ before multiplying out lets you factor $81$ from both terms and read $x=81$ directly, without any large arithmetic.