How do you simplify ${\left( {3{x^3}} \right)^2}$?

Hint: First of all we will take the given expression and will use the law of power and exponent for the product of the power rule and will simplify the expression for the resultant required value.

Complete step-by-step solution:
Take the given expression: ${\left( {3{x^3}} \right)^2}$
Since, the power is applied to the whole bracket therefore power is applied to all the terms inside the bracket.
$ = {3^2} \times {\left( {{x^3}} \right)^2}$
When any exponent is having power two then the number is multiplied with itself twice which we call square of that number in the general form. Also using the law of power and exponent which states that when there is power to the power then both the powers are multiplied in such a way as: ${({x^m})^n} = {x^{mn}}$. Apply in the above expression and simplify.
$ = 9 \times \left( {{x^6}} \right)$
Simplify the above expression.
$ = 9{x^6}$
Hence the simplified form of ${\left( {3{x^3}} \right)^2}$is $9{x^6}$

Note: The power is used to express mathematical equations in the short form; it is an expression that represents the repeated multiplication of the same factor. For example - $2 \times 2 \times 2$ can be expressed as ${2^3}$. Here, the number two is called the base and the exponent represents the number of times the base is used as the factor.

Remember the seven basic rules of the exponent or the laws of exponents to solve these types of questions. Make sure to go through the below mentioned rules, it describes how to solve different types of exponents problems and how to add, subtract, multiply and divide the exponents.
i) Product of powers rule
ii) Quotient of powers rule
iii) Power of a power rule
iv) Power of a product rule
v) Power of a quotient rule
vi) Zero power rule
vii) Negative exponent rule