Vamos resolver a equação dada:

[tex]64^{2x - 3} = 16^{x+2}\\\\\left(2^6 \right)^{2x - 3} = \left(2^4\right)^{x + 2}\\\\2^{12x -18} = 2^{4x + 8}\\\\12x - 18 = 4x + 8\\\\8x = 26\\\\x = \frac{26}{8} = \frac{13}{4}.[/tex]

Verificação:

[tex]64^{2 \cdot \frac{13}{4} - 3} = 64^{\frac{7}{2}} = \sqrt{64^7} = \sqrt{2^{42}} = 2^{21}.[/tex]

[tex]16^{\frac{13}{4}+2} = 16^{\frac{21}{4}} = \sqrt[4]{16^{21}} = \sqrt[4]{2^{84}} = 2^{21}.[/tex]

Conclusão:

[tex]64^{2x -3} = 16^{x + 2}[/tex] para [tex]x = \frac{13}{4}.[/tex]