Question

How do you simplify \[-{{5}^{-2}}\]?

Answer 1

Hint: We are asked to simplify the expression \[-{{5}^{-2}}\]. To solve the given problem we should know some of the exponential properties. The first exponential property, we should know states that \[{{\left( ab \right)}^{m}}={{a}^{m}}{{b}^{m}}\]. The other exponential property that we should know is how to evaluate expressions of form \[{{a}^{-b}}\]. This term is evaluated as \[\dfrac{1}{{{a}^{b}}}\]. We will use these two properties, to simplify the given expression.

Complete step by step solution:
We are asked to simplify the expression \[-{{5}^{-2}}\]. This expression is of the form \[{{a}^{-b}}\]. On comparing with this form, we get \[a=-5\] , and \[b=2\]. We know that expressions of these forms are evaluated as \[\dfrac{1}{{{a}^{b}}}\]. Substituting the values of the a and b, we get
\[\Rightarrow {{\left( -5 \right)}^{-2}}=\dfrac{1}{{{\left( -5 \right)}^{2}}}\]
We know the exponential property \[{{\left( ab \right)}^{m}}={{a}^{m}}{{b}^{m}}\]. For the denominator of the above expression, we have \[a=-1\], \[b=5\], and \[m=2\]. Using the above exponential property, we can express the denominator of the term \[\dfrac{1}{{{\left( -5 \right)}^{2}}}\] as follows,
\[\Rightarrow \dfrac{1}{{{\left( -5 \right)}^{2}}}=\dfrac{1}{{{\left( -1 \right)}^{2}}\times {{5}^{2}}}\]
We know that to find the square of a number, we need to multiply it with itself two times. Thus, the square of \[-1\] is \[-1\times -1\] which equals 1. And, the square of 5 is \[5\times 5\] which equals 25.
Substituting these values in the above expression, we get
\[\begin{align}
  & \Rightarrow \dfrac{1}{{{\left( -1 \right)}^{2}}\times {{5}^{2}}}=\dfrac{1}{1\times 25} \\
 & \Rightarrow \dfrac{1}{25} \\
\end{align}\]

Note:
For these types of problems, we should know some of the special exponents. We should also know how to evaluate expressions of the form \[{{a}^{\dfrac{1}{n}}}\] this expression is simplified as \[\sqrt[n]{a}\], which means the \[{{n}^{th}}\] root of the a. Here, if n is an even integer, we must check if a is positive or not. We have already seen one of them in the above example, which is \[{{a}^{-b}}\] and is evaluated as \[\dfrac{1}{{{a}^{b}}}\].