Question

How do you simplify \[4{{x}^{-1}}\] and write it using only positive exponents?

Answer 1

Hint: These types of problems are pretty straight forward and are very simple to solve. For such types of problems we need to have an in-depth knowledge of the theory of power and indices and we also need to have an idea of factorization and the laws of associativity. We also need to know about the general representation of a number in its power and base form. According to the law of power and indices, the associative property holds great and the product of the numbers or variables to the power is equal to the individual numbers or variables to the same power. We know that any power form of a number can be represented as \[{{a}^{b}}\] . Here, ‘a’ is called the “base” and ‘b’ is called the “index” or “power” or “exponent”.

Complete step by step solution:
Now, we start off with our solution by writing that, we first off apply the associativity property to our given problem sum and it forms,
\[\begin{align}
  & 4{{x}^{-1}} \\
 & \Rightarrow 4\times {{x}^{-1}} \\
\end{align}\]
Now, in this problem we convert the negative power of the bases to positive power by taking the reciprocals. We write it as,
\[\Rightarrow 4\times \dfrac{1}{x}\]
Evaluating the values we write,
\[\Rightarrow \dfrac{4}{x}\]
We can observe from this above equation that the equation cannot be further simplified, hence this is our answer to the problem.

Note:
For such problems, where we need to find the square power or cube power of any number or any variable, we must be very careful with the conversion of negative powers to positive powers by taking the reciprocal. We should always remember that the \[{{a}^{-n}}th\] power is equivalent to \[\dfrac{1}{{{a}^{n}}}th\] power of any number or a variable. We must be very careful while applying the law of indices and the other laws of factorizations and associativity and they should not overlap each other.