Question

One ordered pair (a,b) satisfies the two equations ab^4 = 12$ and a^5 b^5 = 7776. What is the value of a in this ordered pair?

Answer 1

Given equations:
`ab^4 = 12 \ and \  a^5 b^5 = 7776.`

Solving first equation for a, we get

`a = (12)/(b^4)`

Substituting
`a = (12)/(b^4)` in second equation, we get

`({ (12)/(b^4)})^5 b^5` = 7776

`(248832)/(b^(15))=7776`

`\mathrm{Multiply\:both\:sides\:by\:}b^(15)`

`248832=7776b^(15)`

`(7776b^(15))/(7776)=(248832)/(7776)`

`b^(15)=32`

`b=32^{(1)/(15)}`

`=\left(2^5\right)^{(1)/(15)}`

`\left(2^5\right)^{(1)/(15)}=2^{5\cdot (1)/(15)}=\sqrt[3]{2}`

`b=\sqrt[3]{2}`

Plugging
`b=\sqrt[3]{2}` in first eqaution.

`\:a\left(\sqrt[3]{2}\right)^4\:=\:12`

`a\cdot \:2\sqrt[3]{2}=12`

`a=3\cdot \:2^{(2)/(3)}`

`a=3\sqrt[3]{4}`.

Therefore, (a,b) =
`(3\sqrt[3]{4},\sqrt[3]{2}).`