If [tex]8a^{3b}[/tex] = 1 and a > 0, find the value of [tex]a^{2b} - \frac{1}{a^{b} }[/tex]

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LammettHash LammettHash · Answer 1 · 2019-04-01T17:58:07+02:00

Some factoring lets us write

[tex]a^{2b}-\dfrac1{a^b}=a^{2b}-a^{-b}=a^{-b}(a^{3b}-1)[/tex]

Then

[tex]8a^{3b}=1\implies a^{3b}=\dfrac18[/tex]

[tex]a^{2b}-\dfrac1{a^b}=a^{-b}\left(\dfrac18-1\right)=-\dfrac78a^{-b}=-\dfrac7{8a^b}[/tex]

Taking the cube root to solve for [tex]b[/tex], we find

[tex]\sqrt[3]{a^{3b}}=\sqrt[3]{\dfrac18}\implies a^b=\dfrac12[/tex]

so ultimately

[tex]a^{2b}-\dfrac1{a^b}=-\dfrac7{8\cdot\frac12}=-\dfrac74[/tex]

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altavistard altavistard · Answer 2 · 2019-04-01T18:06:11+02:00

Answer:

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

Step-by-step 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

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