Manipulation Of Indices

$\begin{aligned} 4^x & = \sqrt{2^{3y}} \\ 2^{2x} & = 2^{\frac{3y}{2}} \\ \Rightarrow 2x & = \frac{3y}{2} \\ x & = \boxed{\dfrac{3y}{4}} \end{aligned}$

The given equation is $4^x=\sqrt{2^{3y}}$ which can be further written as $4^x=\left( 2^{3y}\right)^{1/2}$ . Using the power rule , the equation becomes $4^x=2^{3y/2}$ . To make the bases equal we can write the equation as $2^{2x}=2^{3y/2}$ . Now as we have the bases equal, the exponents too must be equal for the equation to be correct. Hence we have $2x=\frac{3y}{2} \Rightarrow x=\frac{3y}{4}. \square$

$\large {4^x= \sqrt{2^{3y}}}$

$\large 4^x = \left(2^{3y}\right)^{\frac{1}{2}}$

$\large 4^x = \left(\sqrt{4}\right)^{\frac{3y}{2}}$

$\large 4^x = \left(4^{\frac{1}{2}}\right)^{\frac{3y}{2}}$

$\large 4^x = 4^{\frac{3y}{2} \times \frac{1}{2}}$

$\large 4^x = 4^{\frac{3y}{4}}$

$\large x = \boxed{\dfrac{3y}{4}}$

4^x = √(2^[3y])

4^(2x) = 2^(3y)

2^(4x) = 2^(3y)

The same base

4x = 3y

x = 3y / 4