If 8a^(3b) = 1 and a > 0, find the value of a^(2b) - (1)/(a^(b) )

Question

If
`8a^(3b)` = 1 and a > 0, find the value of
`a^(2b) - (1)/(a^(b) )`

Answer 1

The value of the given expression is 1 - 1 = 0

Explanation:

8a^{3b} = 1 can be rewritten as (8a³)^b = 1 = (8a³)^0, which indicates that b = 0. If b= 0, then 2b = 2(0) = 0.

Then a^(2b) = a^0 = 1, and 1/a^b = 1/1 = 1.

Thus, the value of the given expression is 1 - 1 = 0

Answer 2

Some factoring lets us write

`a^(2b)-\frac1{a^b}=a^(2b)-a^(-b)=a^(-b)(a^(3b)-1)`

Then

`8a^(3b)=1\implies a^(3b)=\frac18`

`a^(2b)-\frac1{a^b}=a^(-b)\left(\frac18-1\right)=-\frac78a^(-b)=-\frac7{8a^b}`

Taking the cube root to solve for
`b`, we find

`\sqrt[3]{a^(3b)}=\sqrt[3]{\frac18}\implies a^b=\frac12`

so ultimately

`a^(2b)-\frac1{a^b}=-\frac7{8\cdot\frac12}=-\frac74`