Simplify the given expression \[\sqrt[5]{{243{a^{10}}{b^5}{c^{10}}}}\]

Hint: Here we need to simplify the given expression. This expression consists of roots and powers. So we will first convert the root in the form of power, then we will use the properties of the exponents to simplify it further. After using the different properties of the exponents and using the different mathematical operations, we will get the required answer.

Complete step-by-step answer:
The given expression is \[\sqrt[5]{{243{a^{10}}{b^5}{c^{10}}}}\]
Now, we will first write the root used in the expression in the form of power.
\[ \Rightarrow \sqrt[5]{{243{a^{10}}{b^5}{c^{10}}}} = {\left( {243{a^{10}}{b^5}{c^{10}}} \right)^{\dfrac{1}{5}}}\]
We know from properties of the exponents that \[{\left( {a \cdot b} \right)^2} = {a^2} \cdot {b^2}\].
Using this property of exponents, we get
\[ \Rightarrow \sqrt[5]{{243{a^{10}}{b^5}{c^{10}}}} = {\left( {243} \right)^{\dfrac{1}{5}}}{a^{10 \times \dfrac{1}{5}}}{b^{5 \times \dfrac{1}{5}}}{c^{10 \times \dfrac{1}{5}}}\]
On simplifying the powers, we get
\[ \Rightarrow \sqrt[5]{{243{a^{10}}{b^5}{c^{10}}}} = {\left( {243} \right)^{\dfrac{1}{5}}}{a^2} \cdot b \cdot {c^2}\]
We know that the factorization of the number 243 is equal to \[3 \times 3 \times 3 \times 3 \times 3\] .
So we can write 243 as \[{3^5}\] .
Now, we will substitute this value in the above equation. Therefore, we get
\[ \Rightarrow \sqrt[5]{{243{a^{10}}{b^5}{c^{10}}}} = {\left( {{3^5}} \right)^{\dfrac{1}{5}}}{a^2} \cdot b \cdot {c^2}\]
We know from the properties of the exponents that if we take the power of the exponents, then the power gets multiplied.
Using this property in the expression, we get
\[ \Rightarrow \sqrt[5]{{243{a^{10}}{b^5}{c^{10}}}} = {3^{5 \times }}^{\dfrac{1}{5}}{a^2} \cdot b \cdot {c^2}\]
On further simplifying the power, we get
\[ \Rightarrow \sqrt[5]{{243{a^{10}}{b^5}{c^{10}}}} = 3 \cdot {a^2} \cdot b \cdot {c^2}\]
Hence, the simplified form of the given expression i.e. \[\sqrt[5]{{243{a^{10}}{b^5}{c^{10}}}}\] is equal to \[3 \cdot {a^2} \cdot b \cdot {c^2}\].

Note: Here, exponents are used to represent the number of times the given number is multiplied with itself. When the exponentials with the same base are multiplied then their power gets added. Similarly if the exponential with the same base is divided by another exponential with the same then their power gets subtracted. This is also known as the addition and subtraction properties of the exponentials.