Question

How do you simplify ${{\left( 2{{x}^{4}}{{y}^{-3}} \right)}^{-1}}$ and write it using only positive exponents?

Answer 1

Hint: Problems of powers can be easily solved using the properties of real number indices. We apply the distributive law of exponents so that the negative index outside the bracket can be taken inside the bracket as the indices of the terms inside the bracket. Now, we convert the terms having negative power or exponents into its reciprocal having positive exponents of the same so that the entire expression only contains positive exponents.

Complete step by step answer:
The expression we are given is
${{\left( 2{{x}^{4}}{{y}^{-3}} \right)}^{-1}}$
To start simplifying the given expression we must omit the bracket first. To remove the bracket from the given expression we apply the distributive law of exponents.
According to the Distributive law of exponents If an exponent acts on a single term in parentheses, we can distribute the exponent over the term. As, an example we can write \[{{\left( {{a}^{x}}{{b}^{y}} \right)}^{z}}={{\left( {{a}^{x}} \right)}^{z}}{{\left( {{b}^{y}} \right)}^{z}}\]
Hence, we apply the above formula of exponents in the given expression as shown below
$\Rightarrow {{2}^{-1}}{{\left( {{x}^{4}} \right)}^{-1}}{{\left( {{y}^{-3}} \right)}^{-1}}$
Now, in the above expression we can see that every term has a power which further has a power. To simplify this term, we apply the Power of a power property of indices.
According to the Power of a power property of indices ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ .
Applying the said property in our expression we get
 $\Rightarrow {{2}^{-1}}\left( {{x}^{4\times \left( -1 \right)}} \right)\left( {{y}^{\left( -3 \right)\times \left( -1 \right)}} \right)$
Further simplifying we get
\[\Rightarrow {{2}^{-1}}\left( {{x}^{-4}} \right)\left( {{y}^{3}} \right)\]
Now to get an expression which only contains only positive exponents we must convert the negative exponents into positive exponents. The terms having negative powers can be converted into terms with positive power by just making it reciprocal as
\[\Rightarrow \dfrac{1}{2}\left( \dfrac{1}{{{x}^{4}}} \right)\left( {{y}^{3}} \right)\]
Further simplifying we get
\[\Rightarrow \dfrac{{{y}^{3}}}{2{{x}^{4}}}\]
Therefore, we can rewrite the given expression as \[\dfrac{{{y}^{3}}}{2{{x}^{4}}}\] which only contains positive exponents.

Note:
While solving problems related to power, we must be careful about applying different properties of indices, as applying any property at an inappropriate place may lead to incorrect solutions. Also, we must be careful with the addition and multiplication of the indices, as we often get confused when to add and when to multiply.