Find the product.

(i) $a^2\times \left(2a^{22}\right)\times \left(4a^{26}\right)$

(ii) $\left(\frac{2}{3}xy\right) \times \left(\frac{-9}{10} x^2{y}^2\right)$

(iii) $\left(\frac{-10}{3} p{q}^3\right)\times \left(\frac{6}{5} p^{3}q\right)$

(iv) $\left(x\right)\times \left(x^2\right)\times \left(x^3\right)\times \left(x^4\right)$

(i) Consider the given expression, $a^2\times \left(2a^{22}\right)\times \left(4a^{26}\right)$.

We know that $a^m\times a^n=a^{m+n}$

Perform multiplication to get the following,

$a^2\times \left(2a^{22}\right)\times \left(4a^{26}\right)=(2\times 4) ( a^2\times a^{22}\times a^{26})$

$=8\times a^{2+22+26 }$

$=8a^{50}$

Hence, the simplification of the given expression is  $8a^{50}$.

(ii) Consider the given expression, $\left(\frac{2}{3}xy\right) \times \left(\frac{-9}{10} x^2{y}^2\right)$.

We know that $a^m\times a^n=a^{m+n}$

Perform multiplication to get the following,

$\left(\frac{2}{3}xy\right) \times \left(\frac{-9}{10} x^2{y}^2\right)=\left(\frac{2}{3}\times \frac{-9}{10}\right)\times \left(xy\times x^2{y}^2\right)$

$=\left(\frac{-3}{5}\right)\times \left(x^{2+1}{y}^{2+1}\right)$

$=\left(\frac{-3}{5}\right)\times \left(x^3{y}^3\right)$

Hence, the simplification of the given expression is  $\left(\frac{-3}{5}\right)\left(x^3{y}^3\right)$.

(iii)Consider the given expression, $\left(\frac{-10}{3} p{q}^3\right)\times \left(\frac{6}{5} p^{3}q\right)$.

We know that $a^m\times a^n=a^{m+n}$

Perform multiplication to get the following,

$\left(\frac{-10}{3} p{q}^3\right)\times \left(\frac{6}{5} p^{3}q\right)=\left(\frac{-10}{3}\times \frac{6}{5}\right)\left( p{q}^3\times p^{3}q\right)$

$=\left(-4\right)\times \left(p^{3+1}{q}^{3+1}\right)$

$=\left(-4\right)\times \left(p^4{q}^4\right)$

Hence, the simplification of the given expression is  $-4p^4{q}^4$.

(iv)Consider the given expression, $\left(x\right)\times \left(x^2\right)\times \left(x^3\right)\times \left(x^4\right)$.

We know that $a^m\times a^n=a^{m+n}$

Perform multiplication to get the following,

$\left(x\right)\times \left(x^2\right)\times \left(x^3\right)\times \left(x^4\right)=x^{1+2+3+4}$

$=x^{10}$

Hence, the simplification of the given expression is  $x^{10}$.