Answer:
[tex]\displaystyle\displaystyle \frac{27^{1/3b}}{9^{1/2a}}=\frac{1}{9}[/tex]
Step-by-step explanation:
We are given that a and b are positive integers such that:
[tex]a-b=2[/tex]
And we want to evaluate:
[tex]\displaystyle \frac{27^{1/3b}}{9^{1/2a}}[/tex]
First, note that 27 = 3³ and that 9 = 3². Therefore:
[tex]\displaystyle =\frac{(3^3)^{1/3b}}{(3^2)^{1/2a}}[/tex]
Simplify:
[tex]=\displaystyle \frac{3^b}{3^a}[/tex]
Using the quotient property of exponents:
[tex]=3^{b-a}[/tex]
From our given equation, we can divide both sides by -1 to acquire:
[tex]-a+b=-2\text{ or } b-a=-2[/tex]
Therefore:
[tex]=3^{-2}[/tex]
Hence, our answer is:
[tex]\displaystyle\displaystyle \frac{27^{1/3b}}{9^{1/2a}} =\frac{1}{3^2}=\frac{1}{9}[/tex]
