NECO 2023 · Paper 1 · Q8

Solve $(243)^{2x + 1} = \dfrac{81^{x - 2}}{3^x}$ .

Worked solution (try it first)

Write everything as powers of 3: $243 = 3^5$ and $81 = 3^4$ .
The left side is $3^{5(2x + 1)} = 3^{10x + 5}$ .
On the right, $81^{x - 2} = 3^{4x - 8}$ , and dividing by $3^x$ takes $x$ off the power: $3^{3x - 8}$ .
The bases match, so the powers are equal: $10x + 5 = 3x - 8$ .
Take $3x$ and 5 from both sides: $7x = -13$ , so $x = -\frac{13}{7}$ , option B.