Evaluate $\dfrac{{{{25}^{\dfrac{5}{2}}} \times {{729}^{\dfrac{1}{3}}}}}{{{{125}^{\dfrac{2}{3}}} \times {{27}^{\dfrac{2}{3}}} \times {8^{\dfrac{4}{3}}}}}$.

Hint: First simplify the lowest form of the and put the power of the numerator then use the power of power rule that is ${\left( {{x^m}} \right)^n} = {x^{mn}}$. Use the different concepts of radical addition and subtraction in order to calculate the values.

Complete step by step solution:
According to the given question $\dfrac{{{{25}^{\dfrac{5}{2}}} \times {{729}^{\dfrac{1}{3}}}}}{{{{125}^{\dfrac{2}{3}}} \times {{27}^{\dfrac{2}{3}}} \times {8^{\dfrac{4}{3}}}}}$.
We can be written as $\dfrac{{{{25}^{\dfrac{5}{2}}} \times \root 3 \of {729} }}{{{{125}^{\dfrac{2}{3}}} \times {{27}^{\dfrac{2}{3}}} \times {8^{\dfrac{4}{3}}}}}$.
First, calculate the numerator of the equation. Then, calculate the denominator of the equation.
Simply the ${25^{\dfrac{5}{2}}} = {\left[ {{{\left( 5 \right)}^2}} \right]^{\dfrac{5}{2}}} \Rightarrow {5^{\dfrac{{2 \times 5}}{2}}} = {5^5}$.
Now simplify the equation is $\dfrac{{{5^5} \times \root 3 \of {729} }}{{{{125}^{\dfrac{2}{3}}} \times {{27}^{\dfrac{2}{3}}} \times {8^{\dfrac{4}{3}}}}}$.
Simplify the ${5^5}$ to 3125.
$\Rightarrow$$\dfrac{{3125 \times \root 3 \of {729} }}{{{{125}^{\dfrac{2}{3}}} \times {{27}^{\dfrac{2}{3}}} \times {8^{\dfrac{4}{3}}}}}$
Now, remove the cube root $\root 3 \of {} $ of 729.
$\Rightarrow$$\dfrac{{3125 \times 9}}{{{{125}^{\dfrac{2}{3}}} \times {{27}^{\dfrac{2}{3}}} \times {8^{\dfrac{4}{3}}}}}$.
Now, calculate the $3125 \times 9$ to 28125.
$\Rightarrow$$\dfrac{{28125}}{{{{125}^{\dfrac{2}{3}}} \times {{27}^{\dfrac{2}{3}}} \times {8^{\dfrac{4}{3}}}}}$
Now, write the 125 as ${5^3}$ ,27 as ${3^3}$and 8 as ${2^3}$.
$\Rightarrow$$\dfrac{{28125}}{{{{\left( {{5^3}} \right)}^{\dfrac{2}{3}}} \times {{\left( {{3^3}} \right)}^{\dfrac{2}{3}}} \times {{\left( {{2^3}} \right)}^{\dfrac{4}{3}}}}}$.
Use the rule of the multiple exponents ${\left( {{x^m}} \right)^n} = {x^{mn}}$.
$\Rightarrow$$\dfrac{{28125}}{{{5^{\dfrac{{3 \times 2}}{3}}} \times {3^{\dfrac{{3 \times 2}}{3}}} \times {2^{\dfrac{{3 \times 4}}{3}}}}} \Rightarrow \dfrac{{28125}}{{{5^2} \times {3^2} \times {2^4}}}$.
Now, simplify the equation.
$\Rightarrow$$\dfrac{{28125}}{{25 \times 9 \times 16}} \Rightarrow \dfrac{{28125}}{{225 \times 16}} \Rightarrow \dfrac{{28125}}{{3600}}$.
Now, simplify the lowest form of the fraction.
$\dfrac{{28125}}{{3600}} \Rightarrow \dfrac{{125}}{{16}}$.

Hence, the final answer is $\dfrac{{125}}{{16}}$.

Note: Exponents and powers are ways used to represent very large numbers or very small numbers in a simplified manner. Basically, power is an expression that shows repeated multiplication of the same number or factor. The value of the exponent is based on the number of times the base is multiplied to itself. $\left[ {{{\left( {{x^m}} \right)}^n} = {x^{mn}}} \right]$.