Question

Find the value of (7 1/9)^-1/2, giving your answer as an exact fraction in its simplest form.

Answer 1

`\sf (3)/(8)`

Explanation:

Exponents:

First convert the mixed fraction to improper fraction.

`\sf 7(1)/(9)=(7*9+1)/(9)=(64)/(9)`

Now 64 = 8*8 = 8² and 9 = 3²

`\sf (7(1)/(9))^{(-1)/(2)}= \left((64)/(9)\right)^{(-1)/(2)}\\`

`= \left((9)/(64)\right)^{(1)/(2)}\\\\=\left((3^2)/(8^2)\right)^{(1)/(2)}\\\\=\left(\left((3)/(8)\right)^2\right)^{(1)/(2)}\\\\= \left((3)/(8)\right)^{2*(1)/(2)}\\\\= (3)/(8)`

Hint:

`\sf \left((a)/(b)\right)^(-m)=\left((b)/(a)\right)^m\\\\(a^m)/(b^m)=\left((a)/(b)\right)^m\\\\(a^m)^n=a^(m*n)`