What is the power of quotient property?

Hint: We will start by writing the definition of quotient, power and understand the concepts. Then, we will write the general formula as \[{{\left( \dfrac{a}{b} \right)}^{x}}=\dfrac{{{a}^{x}}}{{{b}^{x}}}\] . Next we will consider an example and simplify it using this concept.

Complete step by step solution:
When the numerator and denominator are raised to the same power separately then after division power of the quotient obtained will also be the same.
We can say that;
\[{{\left( \dfrac{a}{b} \right)}^{x}}=\dfrac{{{a}^{x}}}{{{b}^{x}}}\]………………….. (i)
For example;
\[{{\left( \dfrac{1}{2} \right)}^{2}}=\dfrac{{{1}^{2}}}{{{2}^{2}}}=\dfrac{1}{4}\]
Now, we can verify this property by taking an example;
\[\Rightarrow {{\left( \dfrac{9}{3} \right)}^{2}}=\dfrac{{{9}^{2}}}{{{3}^{2}}}=\dfrac{81}{9}=9\]
On dividing 81 by 9, we get a result as 9.
Now division will be done ahead of applying power;
\[\Rightarrow {{\left( \dfrac{9}{3} \right)}^{2}}={{\left( 3 \right)}^{2}}=9\]
In both cases the result is the same. Hence, we had verified the power of quotient property.
We can feel like this rule is not that much useful in case of numerical type fractions but when it comes to solving algebraic expressions this rule is very useful.
Let us take an example of an algebraic expression;
\[{{\left( \dfrac{a+b}{2a} \right)}^{2}}\]………………… (ii)
It will be very difficult to simplify this algebraic expression without separating the power in numerator and denominator.
Now using power of quotient property;
Equation (ii) can be written as;
\[\Rightarrow \dfrac{{{\left( a+b \right)}^{2}}}{{{\left( 2a \right)}^{2}}}\]
Applying power, we get;
\[\Rightarrow \dfrac{\left( {{a}^{2}}+{{b}^{2}}+2ab \right)}{4{{a}^{2}}}\]
Hence, we conclude that power of quotient property is very useful for solving algebraic equations.

Note: The best method to solve such questions is to explain the concept and write an example to reinforce our understanding. We should remember that this property can be applied to terms containing different numerator and denominator, as well as containing variables, constants, expressions.