Solve (3x - 24) to the fourth power times 7 cubed equals 2 times 7 to the fourth power.

Step-by-step solution

1. 1

   Divide both sides by the common power of 7. $(3x-24)^4 = \frac{2\cdot 7^4}{7^3} = 2\cdot 7^{4-3} = 2\cdot 7 = 14.$ Cancelling $7^3$ first avoids ever computing $7^4 = 2401$.

2. 2

   Ask whether 14 is a perfect fourth power. The fourth powers of small integers are $1, 16, 81, 256,\ldots$, and $14$ is not among them. So no integer (indeed no rational) value of $3x-24$ works, and within the natural numbers the equation has no solution.

3. 3

   Take fourth roots over the reals. An even root has two real branches: $3x-24 = \pm\sqrt[4]{14} = \pm 1.9343364.$ The other two fourth roots of $14$ are purely imaginary and are discarded for a real answer.

4. 4

   Solve each branch. $3x = 24\pm 1.9343364 \quad\Longrightarrow\quad x = 8\pm\frac{\sqrt[4]{14}}{3},$ giving $x \approx 8.6447788$ or $x \approx 7.3552212$.

5. 5

   Verify one root. With $x=8.6447788$, $3x-24 = 1.9343364$ and $(1.9343364)^4 = 14.0000$; multiplying by $7^3=343$ gives $4802 = 2\cdot2401 = 2\cdot7^4$. Correct.

6. 6

   State the answer at the right level. For a school exercise expecting whole numbers the answer is 'no solution'; over the reals it is the symmetric pair $8\pm\tfrac{\sqrt[4]{14}}{3}$, centred on $x=8$ because $3x-24 = 3(x-8)$.