Simplify $\left(\frac{y^{9}}{27x^{-6}}\right)^{\frac{2}{3}}$, leaving your answer in positive index. Answer ... Given $\dfrac{25}{125^{2-x}} = 5^{y}$, express $x$ in terms of $y$. Answer ...

1. Rewrite the expression inside the bracket using positive powers where possible.

$\left(\dfrac{y^9}{27x^{-6}}\right)^{\frac23}=\left(\dfrac{x^6y^9}{27}\right)^{\frac23}$

2. Apply the power $\frac23$ to each factor and simplify.

$\left(\dfrac{x^6y^9}{27}\right)^{\frac23}=\dfrac{(x^6)^{\frac23}(y^9)^{\frac23}}{27^{\frac23}}=\dfrac{x^4y^6}{9}$

1. Express every number in the equation as a power of $5$.

$\frac{25}{125^{2-x}}=5^y$
Since $25=5^2$ and $125=5^3$,
$\frac{5^2}{(5^3)^{2-x}}=5^y$

2. Apply laws of indices and equate powers to solve for $x$.

$(5^3)^{2-x}=5^{3(2-x)}=5^{6-3x}$
So
$\frac{5^2}{5^{6-3x}}=5^{2-(6-3x)}=5^{3x-4}$
Hence
$5^{3x-4}=5^y$
Equating indices,
$3x-4=y$
$3x=y+4$
$x=\frac{y+4}{3}$