Simplify the expression

${\left(5x^{2}y\right)}^2{\left(2x{y}^{3}z\right)}^3\left(4xyz\right)$

According to this law,when a product is raised to a power, every factor of the product is raised to the power.

In general:${\left(xy\right)}^m=x^m{y}^m$

If a term with a power is itself raised to a power then the powers are multiplied together.

In general:${\left(x^m\right)}^n=x^{m\times n}$

If the two terms have the same base (in this case x) and are to be multiplied together, then their indices are added.

In general:$x^m\times x^n=x^{m+n}$

The given expression is:${\left(5x^{2}y\right)}^2{\left(2x{y}^{3}z\right)}^3\left(4xyz\right)$

First apply the product power law to simplify the given expression.

${\left(5x^{2}y\right)}^2{\left(2x{y}^{3}z\right)}^3\left(4xyz\right)=\left[{\left(5\right)}^2{\left(x^2\right)}^2{\left(y\right)}^2\right]\left[{\left(2\right)}^3{\left(x\right)}^3{\left(y^3\right)}^3\left(z\right){}^3\right]\left(4xyz\right)$

Apply the ‘Law of indices’ for power of index numbers and simplify further.

$$\begin{gathered} {\left(5x^{2}y\right)}^2{\left(2x{y}^{3}z\right)}^3\left(4xyz\right)=\left[{\left(5\right)}^2{\left(x^2\right)}^2{\left(y\right)}^2\right]\left[{\left(2\right)}^3{\left(x\right)}^3{\left(y^3\right)}^3\left(z\right){}^3\right]\left(4xyz\right) \\ =\left[25\left(x^{2\times 2}\right)y^2\right]\left[8x^3\left(y^{3\times 3}\right)z^3\right]\left(4xyz\right) \\ =\left(25x^4{y}^2\right)\left(8x^3{y}^9{z}^3\right)\left(4xyz\right) \end{gathered}$$

Collect the like terms together and use multiplication rule to simplify further.

$$\begin{gathered} {\left(5x^{2}y\right)}^2{\left(2x{y}^{3}z\right)}^3\left(4xyz\right)=\left(25x^4{y}^2\right)\left(8x^3{y}^9{z}^3\right)\left(4xyz\right) \\ =\left(25\times 8\times 4\right)\left(x^4\times x^3\times x\right)\left(y^2\times y^9\times y\right)\left(z^3\times z\right) \\ =800\left(x^{4+3+1}\right)\left(y^{2+9+1}\right)\left(z^{3+1}\right) \\ =800\left(x^8\right)\left(y^{12}\right)\left(z^4\right) \\ =800x^8{y}^{12}{z}^4 \end{gathered}$$