Given equations: [tex]ab^4 = 12 \ and \ a^5 b^5 = 7776.[/tex]
Solving first equation for a, we get
[tex]a = \frac{12}{b^4}[/tex]
Substituting [tex]a = \frac{12}{b^4}[/tex] in second equation, we get
[tex]({ \frac{12}{b^4}})^5 b^5[/tex] = 7776
[tex]\frac{248832}{b^{15}}=7776[/tex]
[tex]\mathrm{Multiply\:both\:sides\:by\:}b^{15}[/tex]
[tex]248832=7776b^{15}[/tex]
[tex]\frac{7776b^{15}}{7776}=\frac{248832}{7776}[/tex]
[tex]b^{15}=32[/tex]
[tex]b=32^{\frac{1}{15}}[/tex]
[tex]=\left(2^5\right)^{\frac{1}{15}}[/tex]
[tex]\left(2^5\right)^{\frac{1}{15}}=2^{5\cdot \frac{1}{15}}=\sqrt[3]{2}[/tex]
[tex]b=\sqrt[3]{2}[/tex]
Plugging [tex]b=\sqrt[3]{2}[/tex] in first eqaution.
[tex]\:a\left(\sqrt[3]{2}\right)^4\:=\:12[/tex]
[tex]a\cdot \:2\sqrt[3]{2}=12[/tex]
[tex]a=3\cdot \:2^{\frac{2}{3}}[/tex]
[tex]a=3\sqrt[3]{4}[/tex].
