Question

How do you simplify $ \dfrac{1}{{{x^{ - 3}}}}? $

Answer 1

Hint: In this problem we have given one fraction and in that fraction the denominator has some value in its power and the power value is a negative value. So here our work is just to simplify the fraction. That is we have to make the given fraction into some integer.

Formula used: Negative exponents: $ {a^{ - n}} = \dfrac{1}{{{a^n}}}{\text{ for }}a \ne 0 $

Complete step-by-step solution:
Given fraction is $ \dfrac{1}{{{x^{ - 3}}}} $
We know that negative exponents means that we have to flip the sign over.
That is $ {x^{ - 1}} = \dfrac{1}{x} $
Therefore, we are going to apply this in our given fraction $ \dfrac{1}{{{x^{ - 3}}}} $ also. Then it becomes $ \dfrac{1}{{\dfrac{1}{{{x^3}}}}} $ .
The negative sign becomes positive when we flip the fraction.
Now we just do fractional division, here we just flip the denominator over and multiply with the numerator.
Then it becomes, $ 1.\dfrac{{{x^3}}}{1} $
Any number divided by one again becomes the same number.
 $ \Rightarrow 1.{x^3} $
If we multiply any number with one then we get the same number and there will be no changes in it.
 $ \Rightarrow {x^3} $

Therefore, the required answer after the simplification is $ {x^3} $ .

Note: The negative sign on an exponent means the reciprocal. If we have given a negative exponential means just apply the negative exponential rule. Negative exponents in the numerator get moved to the denominator and it becomes positive exponents. And negative exponents in the denominator get moved to the numerator and it becomes positive exponents.
The main thing about this problem is the numerator. Before we start to simplify the given fraction we have to confirm that the denominator value is not equal to zero. If the denominator value is equal to zero then we could not continue the simplification process.