Evaluate \[\sqrt[3]{{125 \times 64}}\].

Hint:
Here, we need to simplify the given expression. We will rewrite the expression inside the cube root as a power of 3 using divisibility tests and rules of exponents. Then, we will use rules of exponents to simplify the given expression.

Formula Used:
We will use the following formulas:
1) If two or more numbers with same base and different exponents are multiplied, the product can be written as \[{a^b} \times {a^c} \times {a^d} = {a^{b + c + d}}\].
2) If two or more numbers with different bases and same exponent are multiplied, the product can be written as \[{a^m} \times {b^m} = {\left( {ab} \right)^m}\].
3) If a number with an exponent is raised to another exponent, then the exponents are multiplied. This can be written as \[{\left( {{a^m}} \right)^n} = {a^{m \times n}}\].

Complete step by step solution:
We will rewrite the expression inside the cube root and use rules of exponents to simplify the expression.
First, we will express the numbers 125 and 64 as a power of 3.
Let us use the divisibility tests of 2, 3, 5, etc. to find the factors of 125.
First, we will check the divisibility by 2.
We know that a number is divisible by 2 if it is an even number.
This means that any number that has one of the digits 2, 4, 6, 8, or 0 in the unit’s place, is divisible by 2.
We can observe that the number 125 has 5 at the unit’s place.
Therefore, the number 125 is not divisible by 2.
Next, we will check the divisibility by 3.
A number is divisible by 3 if the sum of its digits is divisible by 3.
We will add the digits of the number 125.
Thus, we get
\[1 + 2 + 5 = 8\]
Since the number 8 is not divisible by 3, the number 125 is not divisible by 3.
Next, we will check the divisibility by 5.
A number is divisible by 5 if it has the digit 0 or 5 at the unit’s place.
We can observe that the number 125 has 5 at the unit’s place.
Therefore, the number 125 is divisible by 5.
Dividing 125 by 5, we get
\[\dfrac{{125}}{5} = 25\]
We know that 25 is the product of 5 and 5.
Therefore, we can rewrite the number 125 as
\[125 = 5 \times 5 \times 5\]
We will use the rule of exponent, \[{a^b} \times {a^c} \times {a^d} = {a^{b + c + d}}\]. Therefore, we can rewrite 125 as
\[\begin{array}{l} \Rightarrow 125 = {5^{1 + 1 + 1}}\\ \Rightarrow 125 = {5^3}\end{array}\]
\[\therefore \] We have expressed 125 as 5 raised to the power 3.
Let us use the divisibility tests of 2, 3, 5, etc. to find the factors of 64.
First, we will check the divisibility by 2.
We can observe that the number 64 has 4 at the unit’s place.
Therefore, the number 64 is divisible by 2.
Dividing 64 by 2, we get
\[\dfrac{{64}}{2} = 32\]
Now, the number 32 has 2 at the unit’s place.
Therefore, the number 32 is divisible by 2.
Dividing 32 by 2, we get
\[\dfrac{{32}}{2} = 16\]
We know that 16 is the product of 4 and 4.
Therefore, we can rewrite the number 64 as
\[64 = 2 \times 2 \times 4 \times 4\]
Multiplying 2 by 2 in the expression, we get
\[64 = 4 \times 4 \times 4\]
Applying the rule of exponent \[{a^b} \times {a^c} \times {a^d} = {a^{b + c + d}}\], we can rewrite 64 as
\[\begin{array}{l} \Rightarrow 64 = {4^{1 + 1 + 1}}\\ \Rightarrow 64 = {4^3}\end{array}\]
\[\therefore \]We have expressed 64 as 4 raised to the power 3.
Finally, we will simplify the given expression.
Substituting \[64 = {4^3}\] and \[64 = {4^3}\] in the expression \[\sqrt[3]{{125 \times 64}}\], we get
\[ \Rightarrow \sqrt[3]{{125 \times 64}} = \sqrt[3]{{{5^3} \times {4^3}}}\]
Rule of exponent: If two or more numbers with different bases and same exponent are multiplied, the product can be written as \[{a^m} \times {b^m} = {\left( {ab} \right)^m}\].
Applying the rule of exponent \[{a^m} \times {b^m} = {\left( {ab} \right)^m}\], we get
\[ \Rightarrow \sqrt[3]{{125 \times 64}} = \sqrt[3]{{{{\left( {5 \times 4} \right)}^3}}}\]
Multiplying the terms, we get
\[ \Rightarrow \sqrt[3]{{125 \times 64}} = \sqrt[3]{{{{20}^3}}}\]
Rewriting the equation, we get
\[ \Rightarrow \sqrt[3]{{125 \times 64}} = {\left( {{{20}^3}} \right)^{\dfrac{1}{3}}}\]
Applying the rule of exponent \[{\left( {{a^m}} \right)^n} = {a^{m \times n}}\], we get
\[ \Rightarrow \sqrt[3]{{125 \times 64}} = {20^{3 \times \dfrac{1}{3}}}\]
Therefore, we get
\[ \Rightarrow \sqrt[3]{{125 \times 64}} = {20^1} = 20\]

Thus, we get the value of the expression \[\sqrt[3]{{125 \times 64}}\] as 20.

Note:
We expressed 125 and 64 as 5 raised to the power 3, and 4 raised to the power 3, respectively. Here, 125 is called the cube of 5, and 64 is called the cube of 4. When a number is raised to the power 3, the resulting product is its cube. On the other hand, cube root of a number gives all the factors which when multiplied give the original number.
We simplified \[{20^1}\] as 20. This is because any number raised to the exponent 1 is equal to itself.