Find x if 3 to the 15th power divided by (125 - 4x) equals 3 to the 13th power.

Step-by-step solution

1. 1

   Do not expand the powers. $3^{15}$ is over $14$ million; the whole point is to keep it symbolic. Read the equation as 'dividend divided by unknown divisor equals quotient'.

2. 2

   Recover the divisor. In $a:b=c$ the divisor is $b = a:c$, so $125-4x = 3^{15}:3^{13}.$

3. 3

   Apply the quotient rule for exponents. $3^{15}:3^{13} = 3^{15-13} = 3^2 = 9,$ so $125-4x = 9$.

4. 4

   Solve the linear equation. $4x = 125-9 = 116 \quad\Longrightarrow\quad x = \frac{116}{4} = 29.$

5. 5

   Check, including that the divisor is nonzero. At $x=29$, $125-4(29) = 125-116 = 9 \neq 0$, and $3^{15}:9 = 3^{15}:3^2 = 3^{13}$ exactly as required.

6. 6

   Note the structural trap. Writing $125-4x = 3^{13}:3^{15} = 3^{-2}$ (inverting the quotient) would give a non-integer answer. The divisor is always dividend divided by quotient, never the other way round.