Question

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

Answer 1

Hint: In order to simplify the exponential function \[{\left( {\dfrac{1}{2}} \right)^{ - 2}}\] by comparing with \[{\left( {\dfrac{a}{b}} \right)^x}\]. The positive constant other than \[1\] raised to a variable exponent is called an exponential function. Solving a function at a given input value evaluates it. Where the growth rate and initial value are known, an exponential model can be found.

Complete step by step solution:
In the given problem,
Let the given exponential function is \[ f(x) = {\left( {\dfrac{1}{2}} \right)^{ - 2}}\]
On comparing the given exponential function with \[f(x) = {\left( {\dfrac{a}{b}} \right)^x}\], we can get
Let us write the function as \[{\left( {\dfrac{a}{b}} \right)^x} = \dfrac{{{a^x}}}{{{b^x}}}\]
We write the given function,
 \[{\left( {\dfrac{1}{2}} \right)^{ - 2}} = \dfrac{{{1^{ - 2}}}}{{{2^{ - 2}}}}\]
Since, the value \[1\] raised to any power is just \[1\] as \[{1^{ - 2}} = 1\]. We can get
 \[\dfrac{1}{{{2^{ - 2}}}} = \dfrac{1}{{\dfrac{1}{{{2^2}}}}}\].Since, \[{2^{ - 2}} = \dfrac{1}{{{2^2}}}\]
By simplify the denominator fraction, we can get
  \[\dfrac{1}{{{2^{ - 2}}}} = {2^2} = 4\]
Therefore, the final answer is \[\dfrac{1}{{{2^{ - 2}}}} = 4\].
So, the correct answer is “4”.

Note: An exponential function is a mathematical function, which is used in many real-world situations. It is mainly used to find the exponential decay or exponential growth or to compute investments, model populations and so on. In this article, you will learn about exponential function formulas, rules, properties, graphs, derivatives and exponential series.