Question

How do you simplify $ {\left( {\dfrac{4}{9}} \right)^{ - 2}} $ ?

Answer 1

Hint: In order to answer this question, we have to first understand the concept of inverse. The inverse of a number or a variable is written as $ {X^{ - 1}} $ and this can also be written as $ \dfrac{1}{X} $ . This way a fraction with a negative power value, can be easily converted to a fraction with positive power.

Complete step-by-step answer:
Given to us is a fraction with negative power value. Now we know that the inverse of a number is equal to its reciprocal. This can be applied to powers of higher values as well.
This means that the value of $ {X^{ - 2}} $ can also be written as $ {\left( {\dfrac{1}{X}} \right)^2} $ which is the reciprocal of the initial value.
Here, the value of X is our given fraction. So we can write this as $ X = \dfrac{4}{9} $
Now we can write the value of $ {X^{ - 2}} $ as $ {\left( {\dfrac{4}{9}} \right)^{ - 2}} $
We already know that $ {X^{ - 2}} = {\left( {\dfrac{1}{X}} \right)^2} $
Let us now substitute the value of X in this equation. We can write this as $ {\left( {\dfrac{4}{9}} \right)^{ - 2}} = {\left( {\dfrac{1}{{\left( {\dfrac{4}{9}} \right)}}} \right)^2} $
On solving this we get the value to be $ {\left( {\dfrac{9}{4}} \right)^2} $ and let us now expand the power to both numerator and denominator. Now the value becomes $ \dfrac{{{9^2}}}{{{4^2}}} $
On further solving we get $ \dfrac{{81}}{{16}} $
Therefore, by the method of inversion the value of the given fraction is $ \dfrac{{81}}{{16}} $
So, the correct answer is “ $ \dfrac{{81}}{{16}} $ ”.

Note: It is to be noted that we can further solve this fractional value and convert it into decimal value. This can be done by dividing the numerator i.e. $ 81 $ by the denominator i.e. $ 16 $ . When this division is done, we get the final decimal value as $ 5.0625 $