Question

How do you simplify $ {\left( {\dfrac{1}{9}} \right)^{ - \dfrac{3}{2}}} $ ?

Answer 1

Hint: We are given a negative power which means that we would first have to take reciprocal of the expression and then continue calculating the value of the expression further. So, then we have to find out the square root of the base number. The result thus obtained will be the answer to the given question.

Complete step-by-step answer:
So, the given question requires us to simplify the value of $ \left( {\dfrac{1}{9}} \right) $ raised to the power $ - \dfrac{3}{2} $ .
So, we have, $ {\left( {\dfrac{1}{9}} \right)^{ - \dfrac{3}{2}}} $
So, taking the reciprocal of the expression as the power to which the base number $ \left( {\dfrac{1}{9}} \right) $ is raised is negative, $ - \dfrac{3}{2} $ . So, we get,
 $ \Rightarrow {\left( 9 \right)^{\dfrac{3}{2}}} $
 $ \dfrac{3}{2} $ as a power signifies that we have to compute the cube of the square root of the base number. Hence, taking square root of $ 36 $ , we get,
 $ \Rightarrow {\left( {\sqrt 9 } \right)^3} $
Now, we know that the value of square root of $ 9 $ is $ 3 $ . Hence, we get,
 $ \Rightarrow {3^3} $
Now, we know that the cube of $ 3 $ is $ 27 $ . Hence, we get,
 $ \Rightarrow 27 $
So, the value of $ {\left( {\dfrac{1}{9}} \right)^{ - \dfrac{3}{2}}} $ can be simplified as $ 27 $ .
So, the correct answer is “27”.

Note: These rules or laws of indices help us to minimize the problems and get the answer in very less time. These powers can be positive and negative but can be moulded according to our convenience while solving the problem. Also note that cube-root, square-root is fractions with 1 as numerator and respective root in denominator.