Question

How do you simplify \[{{8}^{\dfrac{-2}{3}}}\] ?

Answer 1

Hint: As we see, we have to simplify the given exponential term so for this first convert the base in exponential terms too. For this we have to recall the exponential laws then using the law of exponential perform further steps

Formula used:
\[{{({{x}^{a}})}^{b}}={{x}^{ab}}\] , \[{{x}^{-n}}=\dfrac{1}{{{x}^{n}}}\] , \[{{a}^{n}}=a\times a\times ......n\text{ }times\]

Complete step-by-step answer:
Here, in the given question we have to simplify \[{{8}^{\dfrac{-2}{3}}}\]
For this, we first convert the base in exponential form
\[\Rightarrow \] \[8\] can be written as \[2\times 2\times 2={{2}^{3}}\]
Now the above term can be written as
\[\Rightarrow {{({{2}^{3}})}^{\dfrac{-2}{3}}}\]
Using the law of exponential \[{{({{x}^{a}})}^{b}}={{x}^{ab}}\]
\[\Rightarrow {{2}^{3\times \dfrac{-2}{3}}}={{2}^{-2}}\]
Now using another law \[{{x}^{-n}}=\dfrac{1}{{{x}^{n}}}\]
\[\Rightarrow {{2}^{-2}}=\dfrac{1}{{{2}^{2}}}\]
\[\Rightarrow \dfrac{1}{2\times 2}=\dfrac{1}{4}\]
Hence the simplified value is \[\dfrac{1}{4}\]

Note: For exponentiality just remember the law or we can say exponential property and and when we see that any property is applicable then just apply as the exponential form is more neat and far better as it is an easier form.