Answer:
[tex]\sf \dfrac{3}{8}[/tex]
Step-by-step explanation:
Exponents:
First convert the mixed fraction to improper fraction.
[tex]\sf 7\dfrac{1}{9}=\dfrac{7*9+1}{9}=\dfrac{64}{9}[/tex]
Now 64 = 8*8 = 8² and 9 = 3²
[tex]\sf (7\dfrac{1}{9})^{\dfrac{-1}{2}}= \left(\dfrac{64}{9}\right)^{\dfrac{-1}{2}}\\[/tex]
[tex]= \left(\dfrac{9}{64}\right)^{\dfrac{1}{2}}\\\\=\left(\dfrac{3^2}{8^2}\right)^{\dfrac{1}{2}}\\\\=\left(\left(\dfrac{3}{8}\right)^2\right)^{\dfrac{1}{2}}\\\\= \left(\dfrac{3}{8}\right)^{2*\dfrac{1}{2}}\\\\= \dfrac{3}{8}[/tex]
Hint:
[tex]\sf \left(\dfrac{a}{b}\right)^{-m}=\left(\dfrac{b}{a}\right)^m\\\\\dfrac{a^m}{b^m}=\left(\dfrac{a}{b}\right)^m\\\\(a^m)^n=a^{m*n}[/tex]
