The speed of a wave produced in water is given by ν = λ ^ a g ^ b ρ...

The dimensional form of velocity can be written as

$\left[v\right] =$\left[{\lambda}^{a}{g}^{b}{ρ}^{c}\right]

$⇒ \left[{M}^{0}L{T}^{- 1}\right]$ = {\left[L\right]}^{a}{\left[L{T}^{- 2}\right]}^{b}{\left[M{L}^{- 3}\right]}^{c}

Comparing the exponents for $M, L, T$,

$ForM, {M}^{c} = {M}^{0}ForL, {L}^{a + b - 3c} = {L}^{1}ForT, {T}^{- 2b} = {T}^{- 1}$

So,

$c = 0 . . . ( i ) a + b - 3c = 1 . . . ( ii ) - 2b = - 1 . . . ( iii )$

From equations (i),(ii),(iii),

$a =$\dfrac{1}{2}$, b =$\dfrac{1}{2}$, c = 0$