Find x if (25 + x) divided by 3 equals 3 to the 14th power divided by 3 to the 12th power.

Step-by-step solution

1. 1

   Simplify the right-hand side before touching x. Never expand $3^{14} = 4{,}782{,}969$; the quotient rule $\dfrac{a^m}{a^n}=a^{m-n}$ collapses it instantly: $3^{14}:3^{12} = 3^{14-12} = 3^2 = 9.$

2. 2

   Rewrite the equation with the simplified right side. $(25+x):3 = 9.$

3. 3

   Undo the division. Multiply both sides by $3$: $25+x = 9\cdot3 = 27.$

4. 4

   Isolate x. Subtract $25$: $x = 27-25 = 2.$

5. 5

   Verify in the original form. $(25+2):3 = 27:3 = 9$, and $3^{14}:3^{12}=3^2=9$. Both sides agree, so $x=2$ is correct.

6. 6

   Note the general shape. Whenever both sides are powers of the same base, subtract exponents first; the arithmetic that remains is always small, no matter how large the original exponents look.