$(5x^2)^3 * (2x^3)^5 = 2^2 * 10^3$
$125x^6 * 32x^{15} = 4 * 1000$
$125x^6 * 32x^{15} = 4000$
$4000x^{21} = 4000$
$x^{21} = 1$
x = 1

$(9x^4)^2 * (\frac{1}{2}x^2)^8 = (\frac{3}{4})^4$
$81x^8 * \frac{1}{256}x^{16} = \frac{81}{256}$
$\frac{81}{256}x^{24} = \frac{81}{256}$
$x^{24} = 1$
x = ±1

$(3x^3)^4 * (4x^5)^3 = -72^2$
$81x^{12} * 64x^{15} = -5184$
$5184x^{27} = -5184$
$x^{27} = -1$
x = −1

$(8x^5)^2 * (\frac{1}{5}x^4)^3 = (\frac{4}{5})^3$
$64x^{10} * \frac{1}{125}x^{12} = \frac{64}{125}$
$\frac{64}{125}x^{12} = \frac{64}{125}$
$x^{12} = 1$
x = ±1