解方程 $3^{3x} \times 125^{x} = 225$。

1. Express each number as powers of primes.

$125=5^3, \qquad 225=15^2=3^2\times 5^2$
So
$3^{3x}\times 125^x=225$
becomes
$3^{3x}\times (5^3)^x=3^2\times 5^2.$

2. Apply laws of indices to simplify the left-hand side.

$(5^3)^x=5^{3x}$
Hence
$3^{3x}\times 5^{3x}=3^2\times 5^2.$

3. Combine the powers and solve for $x$.

$3^{3x}\times 5^{3x}=(3\times 5)^{3x}=15^{3x}$
and
$3^2\times 5^2=(3\times 5)^2=15^2.$
Thus
$15^{3x}=15^2.$
Equating indices,
$3x=2$
$x=\frac{2}{3}.$