Question

Find the value of $y$ in the expression:
$(m^{\frac{1}{3}}{)}^{\frac{1}{5}}=m^{\frac{1}{y}}$

Answer 1

We need to find the value of $y$ in the polynomial equation:
$(m^{\frac{1}{3}}{)}^{\frac{1}{5}}=m^{\frac{1}{y}}$

$\underset{–}{Step1:}$

First, let's solve the LHS:
$(m^{\frac{1}{3}}{)}^{\frac{1}{5}}=m^{\frac{1}{3}\times \frac{1}{5}}$ (Since $(a^b{)}^c=a^{(b\times c)}$)

$=m^{\frac{1}{15}}$


$\underset{–}{Step2:}$

Now, let's equate the LHS and RHS:

$m^{\frac{1}{15}}=m^{\frac{1}{y}}$

We see in the above equation the bases are same, i.e., $m$, hence, we can say that the exponents are equal. ($\because a^p=a^q$ means that $p=q$)

$\frac{1}{15}=\frac{1}{y}$
$⟹15=y$
$⟹y=15$

The value of $y$ in the polynomial equation $(m^{\frac{1}{3}}{)}^{\frac{1}{5}}=m^{\frac{1}{y}}$ is $15$.