Question

Evaluate the given expression :\[{\left( { - 4} \right)^{ - 2}}\]
A.\[ - \dfrac{1}{{16}}\]
B.\[\dfrac{1}{{16}}\]
C.\[ - 16\]
D.\[16\]

Answer 1

Hint: In the given question, we have been asked to calculate a given expression. To solve the question, we need to know how to convert a negative exponential power to a positive exponential power. We do that, and then we just solve the exponent as normal.

Formula Used:
To solve the question, we are going to use the formula to convert a negative exponential power to a positive exponential power, which is,
\[{a^{ - b}} = \dfrac{1}{{{a^b}}}\]

Complete step-by-step answer:
In the question, the expression to be solved is \[{\left( { - 4} \right)^{ - 2}}\].
First, we convert the negative power to positive,
\[{\left( { - 4} \right)^{ - 2}} = \dfrac{1}{{{{\left( { - 4} \right)}^2}}}\]
Now, we just solve the expression as normal,
\[\dfrac{1}{{{{\left( { - 4} \right)}^2}}} = \dfrac{1}{{ - 4}} \times \dfrac{1}{{ - 4}} = \dfrac{1}{{16}}\]
Hence, the correct option is B.

Additional Information:
So, if the power is negative, we inverse the number and solve the exponent normally. But, if the power is a fraction, then it is a root exponent, for example, \[{a^{\dfrac{1}{3}}} = \sqrt[3]{a}\]. Hence, if we have a number of the form \[{a^{\dfrac{m}{n}}}\], then it can be written as,
\[\sqrt[n]{{{{\left( a \right)}^m}}}\]

Note: The negative power only affects the fraction kind of thing of the number. It does not change anything about the sign with the number. So, if we have a negative power, we just take the reciprocal of the number, and calculate the number normally.