Question

How do you simplify $ - {\left( {\dfrac{1}{2}} \right)^4} $ ?

Answer 1

Hint: We can see this problem is from indices and powers. This number given is having $ \dfrac{1}{2} $ as base and $ 4 $ as power. But we are also given a negative sign in front of the expression consisting of power and exponent. So, we first have to find out the fourth power of the base number, $ \dfrac{1}{2} $ and then multiply the number obtained by $ \left( { - 1} \right) $ . 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 negative of $ \dfrac{1}{2} $ raised to the power $ 4 $ . $ \dfrac{1}{2} $ raised to the power $ 4 $ can be written as $ {\left( {\dfrac{1}{2}} \right)^4} $ .
This is of the form $ {\left( {\dfrac{a}{b}} \right)^m} $ . But we can rewrite the expression using the laws of indices and powers as $ \left( {\dfrac{{{a^m}}}{{{b^m}}}} \right) $ using the property $ {\left( {\dfrac{a}{b}} \right)^m} = \left( {\dfrac{{{a^m}}}{{{b^m}}}} \right) $ .
Thus we will apply the same property in the question as well.
So, we have, $ - {\left( {\dfrac{1}{2}} \right)^4} $
 $ \Rightarrow - \left( {\dfrac{{{1^4}}}{{{2^4}}}} \right) $
Now, we know that the value of $ {2^4} $ is equal to $ 16 $ .
 $ \Rightarrow - \left( {\dfrac{1}{{16}}} \right) $
Opening the bracket and representing the answer in a presentable manner, we get,
 $ \Rightarrow - \dfrac{1}{{16}} $
So, the value of $ - {\left( {\dfrac{1}{2}} \right)^4} $ can be simplified as $ - \dfrac{1}{{16}} $ .
So, the correct answer is “ $ - \dfrac{1}{{16}} $ ”.

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.