Question

How do you simplify $ {{\left( 3{{x}^{4}}{{y}^{5}} \right)}^{-3}} $ ?

Answer 1

Hint: We will look at the definition of exponents. We will consider the case when the exponent is negative. We will look at an example of such a case. We will see some of the rules for simplification of expressions involving exponents. Then we will use these rules, one by one, to simplify the given expression.

Complete step by step answer:
The exponent is defined as the power to which a given number or expression is to be raised. So, in the expression $ {{a}^{b}} $ , the exponent is $ b $ and the base is $ a $ . Now, we will consider the case when the exponent is negative. If we have a negative exponent then it means that we have to take the reciprocal of that number. Or we can say that we have to divide by the base instead of multiplying it. So, if we have $ {{a}^{-b}} $ , then this is equal to $ \dfrac{1}{{{a}^{b}}} $ . For example, consider $ {{2}^{-3}} $ . We have the following,
 $ {{2}^{-3}}=\dfrac{1}{{{2}^{3}}}=\dfrac{1}{8} $
We have some rules of exponents. These are as follows,
(i) $ {{\left( ab \right)}^{n}}={{a}^{n}}{{b}^{n}} $
(ii) $ {{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}} $
Now, the given expression is $ {{\left( 3{{x}^{4}}{{y}^{5}} \right)}^{-3}} $ . Since we have a negative exponent, we can write the given expression as the following,
 $ {{\left( 3{{x}^{4}}{{y}^{5}} \right)}^{-3}}=\dfrac{1}{{{\left( 3{{x}^{4}}{{y}^{5}} \right)}^{3}}} $
Next, we will use rule (i) mentioned above to simplify the above equation,
 $ {{\left( 3{{x}^{4}}{{y}^{5}} \right)}^{-3}}=\dfrac{1}{{{\left( 3 \right)}^{3}}{{\left( {{x}^{4}} \right)}^{3}}{{\left( {{y}^{5}} \right)}^{3}}} $
Now, we will use rule (ii) mentioned above for further simplification in the following manner,
 $ \begin{align}
  & {{\left( 3{{x}^{4}}{{y}^{5}} \right)}^{-3}}=\dfrac{1}{27{{x}^{4\times 3}}{{y}^{5\times 3}}} \\
 & \therefore {{\left( 3{{x}^{4}}{{y}^{5}} \right)}^{-3}}=\dfrac{1}{27{{x}^{12}}{{y}^{15}}} \\
\end{align} $

Note:
 There are more rules involving the exponents. Some of them are $ {{a}^{m}}\times {{a}^{n}}={{a}^{m+n}} $ ; $ {{a}^{\dfrac{m}{n}}}=\sqrt[n]{{{a}^{m}}} $ ; etc. These rules are very useful for simplification of expressions. It makes it easy for us to do the calculations involving exponents. We should be familiar with all the rules involving exponents.