How do you simplify \[{\left( {\dfrac{7}{{12}}} \right)^2}\]?

Hint: To solve this question, we will use the properties of the powers or the exponents. We will apply the given exponent on the number in the denominator and numerator. Then we will simplify it further to get the required value.

Complete step-by-step solution:
The given expression can also be written as
\[{\left( {\dfrac{7}{{12}}} \right)^2} = {\left( {7 \times \dfrac{1}{{12}}} \right)^2}\]
So we have the product raised to a power. By the property of the exponent to the product, we know that we can split the above expression as a product of each number raised to the same power. So the above expression can be written as-
\[\begin{array}{l} \Rightarrow {\left( {\dfrac{7}{{12}}} \right)^2} = {7^2} \times \dfrac{1}{{{{12}^2}}}\\ \Rightarrow {\left( {\dfrac{7}{{12}}} \right)^2} = \dfrac{{{7^2}}}{{{{12}^2}}}\end{array}\]
Now, we know that the square of a number is equal to the number multiplied by itself. So the numerator and the denominator in the above expression is written as-
\[\begin{array}{l} \Rightarrow {\left( {\dfrac{7}{{12}}} \right)^2} = \dfrac{{7 \times 7}}{{12 \times 12}}\\ \Rightarrow {\left( {\dfrac{7}{{12}}} \right)^2} = \dfrac{{49}}{{144}}\end{array}\]

Hence, the given expression is simplified as \[\dfrac{{49}}{{144}}\]

Additional Information:
There are three main types of fractions i.e. proper fractions, improper fractions, and mixed fractions. A proper fraction is a fraction having the numerator less or lowers in degree, than the denominator. The value of proper fraction after simplification is always less than 1. An improper fraction is a fraction where the numerator is greater than or equals to the denominator, then it is known as an improper fraction. A mixed Fraction is the combination of a natural number and fraction.

Note:
We do not necessarily need to apply the rule of the exponent to the product. We can directly consider the square of the fraction given in the question and simplify it to get the same simplified expression, as obtained in the above solution. The property of the exponent of the product states that the power to a given product of terms is equal to the product of each term raised to the same power.