Question

How do you simplify \[{{\left( 2{{x}^{4}}{{y}^{-3}} \right)}^{-1}}\]?

Answer 1

Hint: Exponent is a quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression like \[{{x}^{a}}\]. We use the formulae \[{{\left( {{a}^{m}}{{b}^{n}} \right)}^{p}}={{a}^{mp}}{{b}^{np}}\] and \[{{a}^{-m}}\Leftrightarrow \dfrac{1}{{{a}^{m}}}\] for simplification. The value of an exponent raised to another exponent is \[{{\left( {{a}^{b}} \right)}^{c}}={{a}^{bc}}\].

Complete step by step answer:
As per the given question, we have to simplify the given expression. The given expression consists of exponents. So, we need to use the concept of exponents and the laws of exponents to simplify the given expression. Here, the given expression is \[{{\left( 2{{x}^{4}}{{y}^{-3}} \right)}^{-1}}\].
One way of visualising basic positive integer exponent is as repeated multiplication \[{{a}^{n}}=a\times a\times ...\times a(n-times)\]
Then \[{{\left( {{a}^{b}} \right)}^{c}}\] means \[\left( {{a}^{b}} \right)\] multiplied c times. That is, \[{{\left( {{a}^{b}} \right)}^{c}}=\left( {{a}^{b}} \right)\times \left( {{a}^{b}} \right)\times ..c-times....\times \left( {{a}^{b}} \right)=\left( {{a}^{bc}} \right)\]. And we know that \[{{\left( {{a}^{m}}{{b}^{n}} \right)}^{p}}={{a}^{mp}}{{b}^{np}}\] and for negative powers, \[{{a}^{-m}}=\dfrac{1}{{{a}^{m}}}\].
 So, the given expression can be modified using the above-mentioned formulae.
\[\Rightarrow {{\left( 2{{x}^{4}}{{y}^{-3}} \right)}^{-1}}\]
\[\Rightarrow {{2}^{1\times -1}}{{x}^{4\times -1}}{{y}^{-3\times -1}}\] (\[\because {{\left( {{a}^{m}}{{b}^{n}} \right)}^{p}}={{a}^{mp}}{{b}^{np}}\])
Here, multiplication of -1 with 1 is -1, 4 with -1 is -4 and that of -3 with -1 is 3. On substituting these values into the expression, we can rewrite the expression as
\[\Rightarrow {{2}^{-1}}{{x}^{-4}}{{y}^{3}}\]
\[\Rightarrow \dfrac{1}{{{2}^{1}}}\times \dfrac{1}{{{x}^{4}}}\times \dfrac{{{y}^{3}}}{1}\] \[\left( \because {{a}^{-m}}=\dfrac{1}{{{a}^{m}}} \right)\]
Here, 2 to the power 1 is equal to 2. And, multiplication of these three fractions is equal to the multiplication of numerators by multiplication of denominators. Thus, we can rewrite the expression as
\[\Rightarrow \dfrac{1\times 1\times {{y}^{3}}}{2\times {{x}^{4}}\times 1}\]
Multiplication of 1,1 and \[{{y}^{3}}\] is equal to \[{{y}^{3}}\] and multiplication of 2, \[{{x}^{4}}\] and 1 is equal to \[{{x}^{4}}\]. Then the expression becomes
\[\Rightarrow \dfrac{1\times 1\times {{y}^{3}}}{2\times {{x}^{4}}\times 1}=\dfrac{{{y}^{3}}}{2{{x}^{4}}}\]

\[\therefore \dfrac{{{y}^{3}}}{2{{x}^{4}}}\] is the simplified form of \[{{\left( 2{{x}^{4}}{{y}^{-3}} \right)}^{-1}}\].

Note: In order to solve such a type of problem, we must have enough knowledge on the concept of exponents and laws of exponents. While solving the problems, we have to correctly express the exponents. We need to multiply the exponents or add up the powers wherever necessary. We should avoid calculation mistakes to get the desired results.