The dimensions of length are expressed as Gˣ cʸ hᶻ where G, c and h are the universal gravitational constant, speed of light and Planck's constant respectively, then

To express the dimensions of length in terms of the universal gravitational constant ($G$), the speed of light ($c$), and Planck's constant ($h$), we begin by writing their individual dimensional formulas:

1. Speed of light ($c$):
$\left[c\right]=\left[M^{0}L{T}^{-1}\right]$

2. Planck's constant ($h$):
We know energy $E=h\nu$, where $\nu$ is frequency (with dimensions $\left[T^{-1}\right]$).
$\left[h\right]=\frac{\left[E\right]}{\left[\nu \right]}=\frac{\left[M{L}^2{T}^{-2}\right]}{\left[T^{-1}\right]}=\left[M{L}^2{T}^{-1}\right]$

3. Universal gravitational constant ($G$):
From Newton's law of gravitation, $F=\frac{G{m}_1{m}_2}{r^2}$.
$\left[G\right]=\frac{\left[F\right]\left[r^2\right]}{\left[m^2\right]}=\frac{\left[ML{T}^{-2}\right]\left[L^2\right]}{\left[M^2\right]}=\left[M^{-1}{L}^3{T}^{-2}\right]$

We are given that the dimensions of length ($L$) are expressed as:
$\left[L\right]=\left[G^x{c}^y{h}^z\right]$

Substituting the dimensional formulas of $G$, $c$, and $h$ into the equation:
$\left[M^0{L}^1{T}^0\right]={\left[M^{-1}{L}^3{T}^{-2}\right]}^x{\left[L{T}^{-1}\right]}^y{\left[M{L}^2{T}^{-1}\right]}^z$

Combining the exponents for each fundamental quantity:
$\left[M^0{L}^1{T}^0\right]=\left[M^{-x+z}{L}^{3x+y+2z}{T}^{-2x-y-z}\right]$

By equating the exponents of $M$, $L$, and $T$ on both sides, we get a system of three linear equations:
1) For $M$:
$-x+z=0\Rightarrow x=z$
2) For $T$:
$-2x-y-z=0\Rightarrow 2x+y+z=0$
3) For $L$:
$3x+y+2z=1$

Substituting $z=x$ into equations (2) and (3):
From equation (2):
$2x+y+x=0\Rightarrow y=-3x$
From equation (3):
$3x+y+2x=1\Rightarrow 5x+y=1$

Now, substitute $y=-3x$ into $5x+y=1$:
$5x-3x=1$
$2x=1\Rightarrow x=\frac{1}{2}$

Since $z=x$, we have:
$z=\frac{1}{2}$

Using $y=-3x$:
$y=-3\left(\frac{1}{2}\right)=-\frac{3}{2}$

Therefore, the values of the powers are:
$x=\frac{1}{2}, y=-\frac{3}{2}, z=\frac{1}{2}$